This proof focus on the moments summability properties from the global in time moments propagation estimates (REF ), which in fact it is enough to show it for any \(t\in (0,1]\) .

Estimates for summability of moments
In oder to calculate global in time estimates for the propagation of summability of \(sk\) -moments for any rate \(0\le sk \) .
This task is performed by controlling the partial sums \(E^{n}_s(z, t)= \sum _{k=0}^{n}m_{sk}[f](t)\) , for initial data \(f_0\in L^1_{2sk}\) , with \(sk\ge {\bf {k}_{\text{\tiny mx}}}\ge {\bf {k}_{\text{\tiny mn}}}>2\) , uniformly in time \(t\)  and partial summability index \(n\) .
The way to obtain these estimates entices to invoke the estimates for of Ordinary Differential Inequalities (REF ) to obtain an Ordinary Differential Inequality for the \(n\) -sum \(E^{n}_s(z, t)\) .
Here reader needs to observe the following a delicate point regarding the following two needed properties. The first one is the propagation global \(sk\) -moment estimates needs to be performed under the conditions from of Theorem REF 
and Theorem REF , for any \(sk\ge s{\bf {k}_{\text{\tiny mx}}}\) .
The second one consists on the observation that estimates on an ODI for partial sum of for \(E^{n}_s(z, t)\) , needs also global upper bounds for \(\frac{d}{dt} m_{sk}[f](t)=:m^{\prime }_{sk}[f](t)\)  for any \(0\le sk\le s({\bf {k}_{\text{\tiny mx}}}+1)\) .
To this end, start by setting the ODI for \(E^{n}_s(z, t)\)  adding up to order \(0<k\le n\)  the modified \(sk^{th}\) -moments ODIs obtained now from the \(sk^{th}\) -moments of the Boltzmann Equation from Angular Averaging Lemma II (Lemma REF ), and (REF ), to get
\(m_{sk}^{\prime }[f](t) &\le \left[2^{\frac{3\gamma }{2} -1} C_\Phi \mu _{s}\sum ^{k+1}_{j=0} {k \atopwithdelims ()j}m_{s(k-j)}[f](t)\,m_{sj+\gamma /2}[f](t)\right] \mathbb {I}_{ k <{\bf {k}_{\text{\tiny mx}}}+1} + {\mathcal {B}}_{sk}\,\mathbb {I}_{ k \ge {\bf {k}_{\text{\tiny mx}}}+1} \\&< 2^{\frac{3\gamma }{2} -1} {\bf {k}_{\text{\tiny mx}}}C_\Phi \, 2^{\frac{k}{2} +1} \, m_{1+\frac{\gamma }{2}}[f_0]\, m_{s{\bf {k}_{\text{\tiny mx}}}}[f_0]\mathbb {I}_{ k <{\bf {k}_{\text{\tiny mx}}}+1}+{\tt \bf m}_{1,sk}[f_0] \, \beta _{sk} \,R_{{\mathcal {C}}}(\delta ^{-1}_{sk} ) \, \mathbb {I}_{ k \ge {\bf {k}_{\text{\tiny mx}}}+1}. \nonumber \) 
Hence, for the first property, upper bounds for moment's estimates for \(m^{\prime }_{sk}[f](t)\) ,
one can obtain upper bounds for \(sk^{th}\) -moment derivatives, for any \(1<sk\le sk_0\) , such that \(2< s{\bf {k}_{\text{\tiny mx}}}\le sk_0\) .
However, for any initial \(f_0\in L^1_{2sk}\) , with \(sk\ge s{\bf {k}_{\text{\tiny mx}}}\) , clearly \(m_{sk}[f_0](t)\le m_{s{\bf {k}_{\text{\tiny mx}}}}[f_0](t)\) , for any \(0\le sk<{\bf {k}_{\text{\tiny mx}}}\) , and so an upper bound estimate for \(m^{\prime }_{sk} \)  is need for any \(0\le sk\le s{\bf {k}_{\text{\tiny mx}}}\)  to be calculated as follows.
Therefore, after gathering all these estimates related to the coefficients and moments estimates associated to Lebesgue (or polynomial) moments, conditions are sufficient to study moments summability properties in order to find the tails associated to polynomial moments, denoted by \(m_{sk}:=m_{sk}[f](t)\)  in the sequel.
Starting by fixing \(s\in (0,1]\) , summing side-by-side over \(k = 0,\cdots n\)  the sharp form estimates (REF ) of the \(sk^{th}\) -moment of the collision operator, the following estimate to the time variation of partial sum \(E^{n}_s(z, t)\) , for a choice of \(1<{\bf {k}_{\text{\tiny mx}}}<k_0<n\)  to be specified later, yields
\(\frac{d}{dt} E_{s}^{n} (t) &= \frac{d}{dt}\sum _{k=0}^n m_{s k} \frac{z^k}{k!}= \sum _{k=0}^{k_0} \frac{z^{ k}}{k!} m^{\prime }_{s k} + \sum _{k=k_0+1}^n \frac{z^{ k}}{k!} m^{\prime }_{s k},\nonumber \\& \ \ \le \sum _{k=0}^{k_0} m^{\prime }_{s k} \frac{z^k}{k!} +2^{\frac{3\gamma }{2}-1}C_{\Phi } \sum _{k=k_0+1}^n \frac{z^k}{k!} \mu _{s k} S_{s, k} - \mathcal {A}_{s} \sum _{k=k_0+1}^{n} m_{s k + \gamma /2} \,\frac{z^k}{k!}\\&= \sum _{k=0}^{k_0} m^{\prime }_{s k} \frac{z^k}{k!} +2^{\frac{3\gamma }{2}-1}C_{\Phi } \sum _{k=k_0+1}^n \frac{z^k}{k!} \mu _{s k} S_{s, k}- \mathcal {A}_{s} \left( I^{n} _{s,\gamma }(t,z) - \sum _{k=0}^{k_0} m_{s k + \gamma /2} \,\frac{z^k}{k!} \right) \nonumber , \\&= \sum _{k=0}^{k_0}\left( m^{\prime }_{s k} + \mathcal {A}_{s} \sum _{k=0}^{k_0} m_{s k + \gamma /2} \right) \frac{z^k}{k!} +2^{\frac{3\gamma }{2}-1}C_{\Phi } \sum _{k=k_0+1}^n \frac{z^k}{k!} \mu _{s k} S_{s, k}- \mathcal {A}_{s} I^{n} _{s,\gamma }(t,z) \,\frac{z^k}{k!} \nonumber ,\) 

where now  \(\mathcal {A}_{s}= c_{lb} 2^\frac{\gamma }{2} A_{s}\)  is the renormalized coercive constant, defined in (REF ), as the moment estimates bounds (REF ) are also written in terms of \(\mathcal {A}_{s}\) .
Next we focus on estimates of each terms in the partial sum (REF ), for the case of propagation of initial data given by \(E_{s}^{n} (0)= \sum _{k=0}^{n}m_{sk}[f](0)\) , with all \(m_{sk}[f](0)\)  finite, that can be estimated by means of
combining inequalities (REF ) and (REF ). They are
\(m_{sk}^{\prime }&[f](t) <2^{\frac{\frac{k}{2}+3\gamma }{2} } {\bf {k}_{\text{\tiny mx}}}C_\Phi \, m_{1+\frac{\gamma }{2}}[f_0]\, m_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}}[f_0] \mathbb {I}_{ k <{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}}+{\tt \bf m}_{1,sk}[f_0] \, \beta _{sk} \,R_{{\mathcal {C}}}(\delta ^{-1}_{sk}) \, \mathbb {I}_{ k \ge {\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}}\nonumber \\&< 2^{\frac{\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+3\gamma }{2} } {\bf {k}_{\text{\tiny mx}}}C_\Phi \!\left[ m_{1+\frac{\gamma }{2}}[f_0]\, m_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}} [f_0]\mathbb {I}_{m_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}}[f_0] \ge \mathbb {E}_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}}} \right. \nonumber \\&\left.\qquad \qquad + \mathbb {E}_{c(1+\frac{\gamma }{2})}\mathbb {E}_{c(s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2})} \mathbb {I}_{m_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}}[f_0] < \mathbb {E}_{s{\bf {k}_{\text{\tiny mx}}}+1}}\right] \mathbb {I}_{ k < {\bf {k}_{\text{\tiny mx}}}+1}+{\tt \bf m}_{1,sk}[f_0] \, \beta _{sk} \,R_{{\mathcal {C}}}(\delta ^{-1}_{sk}) \, \mathbb {I}_{ k \ge {\bf {k}_{\text{\tiny mx}}}+1} \nonumber \\&< \left[ 2^{\frac{\frac{{\bf {k}_{\text{\tiny mx}}}}{2} +3\gamma }{2} } {\bf {k}_{\text{\tiny mx}}}C_\Phi \left( m_{1+\frac{\gamma }{2}}[f_0]\, m_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}} [f_0] +R_{\mathbb {E}}(\delta ^{-1}_{s+\frac{\gamma }{2}})\,R_{\mathbb {E}}(\delta ^{-1}_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}})\right) \right] \mathbb {I}_{ k <{\bf {k}_{\text{\tiny mx}}}+1} \nonumber \\&\quad \quad \quad \quad \quad \quad +{\tt \bf m}_{1,sk}[f_0] \, \beta _{sk} \,R_{{\mathcal {C}}}(\delta ^{-1}_{sk} ) \, \mathbb {I}_{ k \ge {\bf {k}_{\text{\tiny mx}}}+1} \nonumber \\&=: \left(2\Vert  b\Vert _{L^1(\mathbb {S}^{d\!-\!1})}\right)^{sk+\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+2\gamma +1}\,{\bf C}^P_k(f_0, {\bf {k}_{\text{\tiny mx}}}, \gamma ) \,R_{{\mathcal {C}}}(\delta ^{-1}_{sk} ),\) 

with the constant
\({\bf C}^P_k(f_0, {\bf {k}_{\text{\tiny mx}}}, \gamma )\!=\! \max \left\lbrace {\bf {k}_{\text{\tiny mx}}}C_\Phi m_{1+\frac{\gamma }{2}}[f_0]\, m_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}} [f_0] \!+\!R_{\mathbb {E}}(\delta ^{-1}_{s+\frac{\gamma }{2})}\,R_{\mathbb {E}}(\delta ^{-1}_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}})\, ;\,{\tt \bf m}_{1,sk}[f_0] \!\right\rbrace .\) 
Consequently, taking any \(k_0> {\bf {k}_{\text{\tiny mx}}}+ 2\) , and setting \(\delta _{sk_0 +\frac{\gamma }{2}}= \mathcal {A}_{s}/ \beta _{sk_0+ \frac{\gamma }{2}}\) , the first term in that last RHS of (REF ) is estimated by
\(\sum _{k=0}^{k_0} \left( m^{\prime }_{s k}(t) + \mathcal {A}_{s} m_{s k + \frac{\gamma }{2}} \right) \frac{z^k}{k!}&<k_0\left[ 2^{\frac{k_0 + 3\gamma }{2}} {\bf C}^P_{k_0}(f_0, {\bf {k}_{\text{\tiny mx}}}, \gamma ) + \mathcal {A}_{s}\right] \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ) \sum _{k=0}^{k_0} \frac{z^k}{k!} \nonumber \\& < \kappa ^P_{sk_0}\, \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ) \, e^z\ \le \ \kappa ^P_{sk_0}\, \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ),\) 

for the rate factor \( \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} )=\mathcal {R}(\beta _{sk_0+ \frac{\gamma }{2}}/\mathcal {A}_{s}) \)  as defined in (REF ),
with the propagation factor
\(\kappa ^P_{sk_0}=k_0 \left[\left(2\, \Vert  b\Vert _{L^1(\mathbb {S}^{d\!-\!1})} \right)^{sk+\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+2\gamma +1} {\bf C}^P_{k_0}(f_0, {\bf {k}_{\text{\tiny mx}}}, \gamma ) +\mathcal {A}_{s}\right], \ \) 

with the constant rate \(\mathcal {R}(\beta _{sk_0+ \frac{\gamma }{2}}/\mathcal {A}_{s})\) 
is calculated from (REF ), provided that \(k_0>{\bf {k}_{\text{\tiny mx}}}+1\) , with \({\bf C}^P_{k_0}(f_0, {\bf {k}_{\text{\tiny mx}}}, \gamma )\)  depending on \({\bf {k}_{\text{\tiny mx}}}, \gamma \)  and on the initial \(m_{sk_0+\frac{\gamma }{2}}[f_0]\)  moments, and exponential rate \(z\)  chosen to be
\(z\le \ln e .\) 
The second term from the right hand side estimate (REF ) is controlled by the inequality from the Discrete Convolution Lemma REF  and the fact that \(\mu _{sk}\)  decreases in \(sk\) , to obtain
\(2^{\frac{k_0 + 3\gamma }{2}}C_{\Phi }\sum _{k=k_0+1}^n \frac{z^k}{k!} \mu _{s k_0}S_{s, k} \le 2^{\frac{k_0 + 3\gamma }{2}}C_{\Phi } \mu _{s k}\,E^{n}_{s}(t,z)\,I^{n}_{s,\gamma }(t,z)\,,\qquad sk_0\ge s{\bf {k}_{\text{\tiny mx}}}>1.\) 
Therefore, gathering estimates (REF ) and (REF )
the right hand side
of (REF ) is estimated by
\(\frac{d}{dt} E_{s}^{n} (t,z) \le \kappa ^P_{sk_0}\, \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ) - \Big ( \mathcal {A}_{s} - 2^{\frac{3\gamma }{2}-1}C_{\Phi }\mu _{s k_0}\,E^{n}_{s}(t,z)\Big ) I^{n} _{s,\gamma }(t,z)\,,\qquad sk_0\ge s{\bf {k}_{\text{\tiny mx}}}>1 \,. \) 
Next, step is motivated by the search of an exponential rate \(z\)  small enough that enables an upper bound for \(E^{n}_{s}(t,z)\) . To this end, fix \(z \in (0,z_0]\) , where \(z_0\)  is the exponential rate on the initial data set in (REF ),
and introduce the set of times
\(t_n:=\sup \big \lbrace  t\ge 0\,:\, E^{n}_{s}(\tau ,z) \le M_P + (e-1) m_0[f_0], \; \forall \, \tau \in [0,t] \big \rbrace \,,\quad \text{and}\ \ \ n\in \mathbb {N}.\) 
where \(M_P\)  is the initial moment with exponential tail rate \(z_0\)  and the factor \((e-1)\)  proportional to the initial mass \(m_0[f_0]\)  is the smallest one for any the initial exponential moment with rate \(z_0\)  and order \(s\) , as it will become a transparent choice in the developing estimates.
Clearly, the value of \(0<z_0\)  need to be chosen sufficiently small after the choice of rate (REF ), making \(z\le \min \lbrace  z_0, \ln e\rbrace \le 1\) , for each \(n\)  fixed. Hence the partial moment sum \(E^{n}_{s}(t,z)\)  is uniformly bounded in \(t\)  and \(n\) , by showing \(E^{n}_{s}(t,z) \le M_P + (e-1) m_0[f_0]\)  for an unbounded \(t \in \mathbb {R}^+\) .
Indeed, first note that when \(0 < z < z_0\) ,
\(E^{ n}_{s}(0,z) = \sum _{k=0}^n \frac{z^{ k}}{k!} m_{s k}(0) \le E_{s}({z_0},0)= M_P,\quad \text{uniformly in} \ n\in \mathbb {N}, \ \text{for any} \ z\le \min \lbrace  z_0, \ln e\rbrace .\) 
Now, since each \(m_sk[f]\) (t) moment is continuous in time \(t\) , then the partial sum \(E_{s}^n(t,z)\)  is also continuous in \(t\) . Hence, since \(E_{s}^n(t, z)\le M_P + (e-1)m_0[f_0]\) , for \(t\in [0,t_n]\) , then the right hand side of inequality (REF ) is dominated by
\(\frac{d}{dt} E_{s}^{ n} (t) &\le \kappa ^P_{sk_0} \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ) \!- \! \left( \mathcal {A}_{s} \!- \! 2^{\frac{3\gamma }{2}-1}C_{\Phi }\mu _{s k_0} (M_P \! + \! (e \!- \!1) m_0[f_0])\right)I^{ n} _{s,\gamma }(t,z) , \) 

for any \(z\le \min \lbrace  z_0, \ln e\rbrace \) , with rate \(\mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} )\)  from (REF ) and \( sk_0 > s {\bf {k}_{\text{\tiny mx}}}> 2\) .
Therefore, since \(0<\mu _{sk} \searrow 0\)  as \(k\rightarrow 0\) , then it is possible to choose a sufficiently large \(k_0\) , cutting-off in the initial partial sums in (REF ), such that
\(&\mathcal {A}_{s}- 2^{\frac{3\gamma }{2}-1}C_{\Phi } \mu _{s k_{0}}(M_P+(e-1)m_0[f_0]) \ge \frac{ \mathcal {A}_{s}}{2}, \ \ \text{or, equivalently,} \nonumber \\&\qquad \qquad \qquad \qquad \mu _{s k_{0}} \ge \frac{\mathcal {A}_{s}}{2^{\frac{3\gamma }{2}} C_{\Phi } (M_P+ (e-1)m_0[f_0])}.\) 
Remark 18  Note that if the angular transition \(b(\hat{u}\cdot \sigma ) \in L^p(S^{d-1})\) ,  for \(1<p\le \infty \) , then by (REF ), the \(k_0\)  moment order can be chosen by taking
\(sk_0= \mathcal {O}\left(\frac{2^{\frac{3\gamma }{2}} C_{\Phi } (M_P+ (e-1)m_0[f_0])}{\mathcal {A}_{s}}\right)^q, \qquad \text{for} \ q= \frac{p}{p-1} \ \ \text{and} \ \ \ sk_0>s{\bf {k}_{\text{\tiny mx}}}.\) 

These estimates immediately lead to Ordinary Differential Inequality
\(\frac{d}{dt} E_{s}^n (t, z) \le \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ) \, \kappa ^P_{sk_0} - \frac{\mathcal {A}_{s}}{2} \, I^n _{s,\gamma }(t,z)\,,\qquad \ sk_0>s{\bf {k}_{\text{\tiny mx}}}.\) 
In addition, a lower bound for the shifted moments' partial sum follows form a modified estimate [1]} is
\(I^{n} _{s,\gamma } (t,z) &\ge \sum ^{n}_{k=0}\frac{z^k}{k!}\int _{\lbrace \langle v \rangle \ge z^{-1/2}\rbrace } f(v,t)\langle v \rangle ^{2(sk + \gamma /2)}\text{d}v \nonumber \\& \ge z^{-\gamma /2}\bigg ( E^{n}_{s}(t,z) - \sum ^{n}_{k=0}\frac{z^k}{k!}\int _{\lbrace \langle v \rangle \le z^{-1/2}\rbrace } f(v,t)\langle v \rangle ^{2sk}\text{d}v \bigg )\nonumber \\&\ge z^{-\gamma /2}\bigg ( E^{n}_{s}(t,z) - m_0[f_0]\sum ^{n}_{k=0}\frac{z^{k(1-s)}}{k!} \bigg ) > z^{-\gamma /2} \Big ( E^{n} _{s} (t,z) - e^{z^{1-s}}\,m_0[f_0] \Big ) \\& \ge z^{-\frac{\gamma }{2}} \left(E_{s}^{n}(t,z) - e m_0[f_0] \right)\, . \nonumber \) 

And, since the rate is already chosen to be \(0<z\le \min \lbrace  z_0, \ln e\rbrace ,\)  then \( 1< e^z< e^{z^{1-s}} \le e\)  in the range of \(0<s\le 1\)  values, as the initial data may be an arbitrary rate \(z_0>0.\)
The ODI solution estimate for partial the sums \(E_{s}^{n}(t,z)\)  is finalized after defining the state variable
\(X^{n}_{k_0}(t,z):= E_{s}^{n}(t,z) - e\,m_0[f_0],\) 
whose initial condition is \(X^{n}_{k_0}(0,z_0):= E_{s}^n(0,z_0) - e\,m_0[f_0] = M_P - e\,m_0[f_0].\)  Therefore, writing an new ODI for \(X^n(t,k)\)  after invoking the lower estimate for \(I^{n} _{s,\gamma } (t,z)\)  from (REF ), applied into (REF ) yields
\(\frac{d}{dt}X^{n}_{k_0}(t,z) <-\tfrac{\mathcal {A}_{s}}{2z^{\gamma /2}} \, X^{n}_{k_{0}}(t,z) + \kappa ^P_{sk_0} \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ) \qquad t\in (0,t_{n})\,,\) 
whose time integration of this differential inequality leads to
\(X^{n}_{k_0}(t,z) < X^{n}_{k_0}(0,z)\,e^{-\tfrac{\mathcal {A}_{s}\,t}{2z^{\gamma /2}}} + \frac{2z^{\gamma /2} \kappa ^P_{sk_0} \mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} )}{\mathcal {A}_{s}} \Big (1 - e^{-\tfrac{\mathcal {A}_{s}\,t}{2z^{\gamma /2}}}\Big ),\qquad t\in (0,t_{n})\,.\) 
Thus, the choice of rate \(z>0\)  small enough can be made setting
\(z^{\gamma /2} \, \frac{2\mathcal {R}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ) \, \kappa ^P_{sk_0} }{\mathcal {A}_{s}} = M_P - m_0[f_0] >0,\qquad t\in (0,t_{n})\,,\) 
since \(M_P - m_0[f_0] = \sum _{k=1}^n \frac{z_0^{ k}}{k!} m_{s k}(0)>0\) , or equivalently, identifying a lower bound rate \(z_1\)  by
\(z\le z_1 := \, \left( \frac{ {\mathcal {A}}_{s} \mathcal {R}(\mathcal {A}_{s}/\beta _{sk_0+ \frac{\gamma }{2}}) (M_P-m_0[f_0])}{ \kappa ^P_{sk_0} } \right)^{2/\gamma },\qquad t\in (0,t_{n})\,,\) 
with the rate factor \(\mathcal {R}(\mathcal {A}_{s}/\beta _{sk_0+ \frac{\gamma }{2}}):= \mathcal {R}(\delta _{sk_0 +\frac{\gamma }{2}} ) =\mathcal {R}^{-1}(\delta ^{-1}_{sk_0 +\frac{\gamma }{2}} ) <1\)  as shown in (REF ), with \(\beta _{sk_0+\frac{\gamma }{2}}\)  calculated from (REF ).
As a consequence the parameter rate \(z_1\)  is uniform in \(n\in {\mathbb {N}}\)  and time \(t\) .
It is also worth to notice that the propagation constant \(\kappa ^P_{sk_0}\) , calculated (REF ) is \(2\kappa ^P_{sk_0}=2^{O({\frac{k_0 + 3\gamma }{2}} +1)},\) 
as a consequence a quantitative estimate for the \(z_1\)  rate can be obtained as follows
\(z_1 &:= 2\left(\mathcal {A}_{s} (M_P-m_0[f_0])\,\frac{\mathcal {R}(\delta _{sk_0 +\frac{\gamma }{2}} )}{\kappa ^P_{sk_0}}\right)^{\frac{2}{\gamma }},\qquad t\in (0,t_{n})\,,\) 

with
\(\left(\frac{ \mathcal {R}(\delta _{sk_0 +\frac{\gamma }{2}} )}{\kappa ^P_{sk_0} }\right)^{\frac{2}{\gamma }}\! =\! \mathcal {O} \left(\frac{1}{2^{\frac{k_0 +3\gamma + 2}{\gamma }} }\right)\left[ \! \left(\delta _{sk_0 +\frac{\gamma }{2}} \right)^{ e_\alpha (sk_0) } \! \! \mathbb {I}_{ k\ge }\! +\! \left(\delta _{sk_0 +\frac{\gamma }{2}}\right)^{\bar{e}_\theta (sk_0) } \! \! \mathbb {I}_{ k < } \right]^{\frac{2}{\gamma }} \! <\! 1,\) 

for \(e_\alpha (sk_0) =c^{-1}_\gamma (sk_0)\)  and \(\bar{e}_\theta (sk_0)\)  defined in (REF ).
Clearly \(z_1\)  degenerates the choice moment rate as \(sk_0\)  grows, so the best possible rate should be taking with the smallest \(sk_0\)  satisfying condition (REF ), or (REF ), depending on the integrabitlity order of the angular functions \(b(\hat{u}\cdot \sigma )\) .
In particular, \(X^{n}_{k_0}(0,z) < E_{s}^{n}(0,z_0) - m_0[f_0]\) , which implies together with (REF ) and (REF ) that
\(E_{s}^{n}(t,z) -e\,m_0[f_0] < M_P - m_0[f_0] \,, \quad \text{for any} \ \ t\in (0,t_{n})\,.\) 
with the upper bound uniform in \(n\in {\mathbb {N}}\) ,or equivalently,
\(E_{s}^{n}(t,z) < M_P + (e-1) m_0[f_0] \qquad \text{for all}\ t\in (0,t_{n})\ \ \text{and} \ z\le \min \lbrace z_0, \ln 2, z_1 \rbrace  <1,\) 

for any \(k> k_0\) , with \(k_0\)  large enough to satisfy condition (REF ), and a uniform in time exponential rate \(z\)  the minimum between \(z_0\)  is the initial exponential rate, \(z_1\)  from condition (REF ) and condition (REF ).
Now, the continuity of \(E_{s}^{n}(z,t)\)  with respect to time \(t\)  ensures that a strict inequality holds, at least, in a slightly larger time interval \([0, t_n + \varepsilon )\) , \(\varepsilon >0\) . Yet, this fact contradicts the maximality of \(t_n\)  unless \(t_n = + \infty \) . Therefore,
\(E_{s}^{n}(z, t) < M_P + (e-1)\,m_0[f_0]\)  for all \(t\ge 0\)  and \(n\in \mathbb {N}\) .
Thus, letting \(n\rightarrow \infty \)  we conclude
\(E_s^{n}[f](z, t) &= \lim \limits _{n\rightarrow \infty }E_{s}^{n}[f](z, t) \le (e-1) m_0[f_0] + M_P \\&\qquad < (e-1)m_0[f_0] + \int _{\mathbb {R}^{d}} f_0(v) \, e^{z_0 \left\langle v \right\rangle ^{2s}} dv = (e-1)m_0[f_0] +E_s[f](z_0, 0), \quad \forall t\ge 0,\) 

i.e. the solution \(f(v,t)\)  to Boltzmann equation with finite initial exponential moment of order \(2s\)  and rate \(z_0\)  propagates the exponential moments of order \(2s\) , but at a lower constant rate \(z\)  than the initial rate \(z_0\)  , with \(z\)  defined by
\(z(t):= z <\min \left\lbrace  z_0,\, \ln e, \, z_1 \right\rbrace ,\) 
for \(z_1\)  defined by (REF ), and well characterized by (REF ).
(Generation of Exponential moments)
This proof, has similarities with respect to the one of previous Theorem REF , mostly in the sense that we seek an ODI for the exponential moment, denoted for simplicity of notation, by \(E_{s}[f](t) := E_{z,s}[f](t) \)  where the exponential order factor \(2s\)  and rate \(z(t)\)  need to be found and fully characterized, by
the initial data \(f_{in}\in L^1_{k_{in}}(\mathbb {R}^{d})\)  for \(k_{in}\ge 2\epsilon +1\) , by the potential rate \(\gamma \in (0,2]\)  and and by properties on the binary collision operator as described for Cauchy Problem stated in Theorem REF . That means the initial data is defined takes the form \(\int _{\mathbb {R}^{d}} f_0(v) \, \left\langle v \right\rangle ^{k_{in}} \mathrm {d}v =: M_G \)  finite, with \(k_{in}\ge > 1\) , as defined in (REF ).
We require \(z(t)\)  to be continuous, increasing for \(t \in (0,1]\)  and with \(z(0)=0\)  since the exponential weight is not available at \(t=0\) .
As the proof for propagation of exponential tails was initiated, in Theorem REF , recall the first the \(sk^{th}\) -moments generation global bounds derived in Theorem REF , (REF ), which can be large for short times.
Set constant \(C_{\gamma k_0} > 0\)  defined by the upper bound of first \(sk_0\)  finite moments, and invoke the \(sk\) -parameters introduced in the polynomial moments estimates, to obtain, for any \(0<s\le 1<s<sk.\) 
Then, starting from the solution of the Boltzmann flow for \(f_{in}(v) \) , constructed in Theorem REF , with the solution set \(\Omega \)  as defined in (REF ),
our task consists on studying the time evolution of generated moment partial power series in \(z(t)\)  up to order \(n\) , denoted \(E^{n}_{s}(z(t),t)\) , with the goal to find uniform in \(n\)  rates \(z(t)>0\)  dependent by the given initial data.
As performed in the proof of the propagation of exponential tail estimates, the corresponding moments ODI satisfied by the solution \(f\)  of the Cauchy problem posed in Theorem REF  written as in (REF ) becomes again the starting point and the representation of \(I^n _{s,\gamma }[f]\)  defined explicitly in (REF ), is now invoked evaluated at \((z(t),t)\)  to obtain
\(I^n _{s,\gamma }[f](z(t), t) := \sum _{k=0}^n m_{sk+\frac{\gamma }{2}}[f](t) \frac{z^k(t)}{k!}\,.\) 
Thus, starting first for values of \(t\in (0,1]\)  and invoking the product rule and the fact that contractive constant \( \mu _k\)  decreasing in \(k\) , \( I^n _{s,\gamma }(z(t),t)\)  given now by the \(n^{th}\) -partial sum of the shifted to the \(sk+\gamma /2\) -moment defined in (REF ), after some elementary algebra it follows that for \(t\in (0,1]\) .
Hence, taking the rate of change of the partial moment's sums
\( E^{n}_{s}(z(t),t)\) , with respect to time \(t\) , and invoking the \(n^{th}\) -partial \(sk^{th}\) -Lesbegue moments sums and the more accurate binary expansion for moments estimates from the collisional form (REF ), the for a suitable \(k_0 > {\bf {k}_{\text{\tiny mx}}}\)  to be chosen later, the following estimate holds
\(\frac{d}{dt} E^{n}_{s}(z(t),t) &= \sum _{k=0}^{k_{0}} m^{\prime }_{sk}[f] \frac{z^k(t)}{k!} + \sum _{k=k_0+1}^n m^{\prime }_{sk}[f] \frac{z^k(t)}{k!} +z^{\prime }(t)\sum _{k=1}^{n} m_{sk}[f] \frac{z^{k-1}(t)}{(k-1)!}\nonumber \\&\le \sum _{k=0}^{k_{0}} m^{\prime }_{sk}[f] \frac{z^k(t)}{k!} + \sum _{k=k_0+1}^n \frac{z^k(t)}{k!} \Big (2^{\frac{3\gamma }{2}-1}C_{\Phi } \mu _{sk} S_{sk}[f] -\mathcal {A}_{s} m_{sk+\frac{\gamma }{2}}[f] \Big ) \nonumber \\&\hspace{122.34692pt} + z^{\prime }(t) \sum _{k=0}^{n-1} m_{sk+\frac{\gamma }{2}}[f] \, \frac{z^k(t)}{(k)!}\\&\le \sum _{k=0}^{k_{0}} \left(m^{\prime }_{sk}[f] + \mathcal {A}_{s} m_{sk+\frac{\gamma }{2}}[f] \right) \frac{z^k(t)}{k!} + 2^{\frac{3\gamma }{2}-1}C_{\Phi }\mu _{sk}\sum _{k=k_0+1}^n \frac{z^k(t)}{k!} S_{sk}[f] \nonumber \\&\hspace{42.67912pt} - \mathcal {A}_{s} I^n _{s,\gamma }(z(t),t) +z^{\prime }(t) I^{n-1}_{s,\gamma }(z(t),t), \nonumber \) 

much like the split made in the Propagation of partial some case (REF ), and so the right hand side from (REF ) will be estimated for times adapting the estimates performed in (REF ), taking into account that the unknown exponetial rate \(z=z(t)\)  depends on time \(t\in (0,1]\) .
Hence, invoking the global moments generation estimates from (REF ),
\(m_{ sk}[f](t) &\le \mathbb {E}_{sk} + m_1[f_0]\bigg ( \frac{2(sk-1)}{\gamma \, \mathcal {A}_{s}\, t}\bigg )^{\frac{2}{\gamma }(sk-1)} \nonumber \\&= \max \left( m_1\,{\tt \bf m}_{1,k}\right)^\frac{ks-1}{sk-1+\frac{\gamma }{2}}[f_0] \, R_{\mathbb {E}}(\delta ^{-1}_{sk}) +m_1[f_0]\bigg ( \frac{1}{c_\gamma (sk)\, \mathcal {A}_{s}\, t}\bigg )^{\frac{1}{c_\gamma (sk-1)}},\) 

as \(c_{\gamma }(sk) = \frac{\gamma }{2(sk-1)}\) , for any \(sk\ge s{\bf {k}_{\text{\tiny mx}}}>2\) ,
the first step starting from consists in performing the \(m^{\prime }_{sk}[f]\)  estimates similar to those on (REF ), for any \(0<sk \le s{\bf {k}_{\text{\tiny mx}}}\)  but now applying the generation upper bounds (REF ) when needed.
In addition, if the initial data \(M_G =m_{s{\bf {k}_{\text{\tiny mx}}}}[f_i](0)\) , and \(k_0\ge \) , then
\(m_{sk}^{\prime }[f](t)&<\! {\tt \bf m}_{1,sk}[f_0] \, \beta _{sk} \,R_{{\mathcal {C}}}(\delta ^{-1}_{sk}) \mathbb {I}_{ k \ge {\bf {k}_{\text{\tiny mx}}}+1}\ +\ 2^{\frac{\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+3\gamma }{2} } {\bf {k}_{\text{\tiny mx}}}C_\Phi m_{1+\frac{\gamma }{2}}[f](t) m_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2}}[f](t) \mathbb {I}_{ k <{\bf {k}_{\text{\tiny mx}}}+1}\nonumber \\&<{\tt \bf m}_{1,sk}[f_0] \, \beta _{sk} \,R_{{\mathcal {C}}}(\delta ^{-1}_{sk}) \, \mathbb {I}_{ k \ge {\bf {k}_{\text{\tiny mx}}}+1}+ 2^{\frac{\frac{{\bf {k}_{\text{\tiny mx}}}}{2} +3\gamma }{2} } \! {\bf {k}_{\text{\tiny mx}}}C_\Phi \left[\left(R_{\mathbb {E}}(\delta ^{-1}_{s+\frac{\gamma }{2} }) + \frac{m_1[f_0]}{\gamma \, A_{s}} \nonumber \right)\right.\\&\left.\quad \quad \left(R_{\mathbb {E}}(\delta ^{-1}_{s{\bf {k}_{\text{\tiny mx}}}+\frac{\gamma }{2} }) \!+ \! \! m_1[f_0]\max _{k<{\bf {k}_{\text{\tiny mx}}}+1}\!\!\left( \!\frac{1}{c_\gamma (sk\!+\!\frac{\gamma }{2}) \mathcal {A}_{s}}\right)^{\!\!\frac{1}{c_\gamma (sk\!+\!\frac{\gamma }{2})}}\right) t^{ -\frac{1}{c_\gamma (sk\!+\!\frac{\gamma }{2})}} \right] \, \mathbb {I}_{ k <{\bf {k}_{\text{\tiny mx}}}+1} \nonumber \\&\ \ < \left(2\, \Vert  b\Vert _{L^1(\mathbb {S}^{d-1})} \right)^{sk+\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+1+2\gamma }\left( {\tt \bf m}_{1,sk}[f_0] \, R_{{\mathcal {C}}} (\delta ^{-1}_{sk}) +\!{\bf C}^G_{\bar{k}}(f_0, C_\Phi ,\gamma ) t^{-\frac{2sk\!+\!\gamma -2}{\gamma }}\right). \) 
Next, the following constant, denoted by \( {\bf C}^G_{\bar{k}}\) , is defined by gathering constant factor depending on moment's oder \(1<k\le \bar{k}\)  such that \( \bar{k}\ge sk> 1\) . This condition readily implies the exponent \(c_\gamma (sk+\frac{\gamma }{2}) >0\) , for any \(sk>1\) . This constant also depends on the initial data \(f_0\)  through
mass and energy, resulting in the following positive and finite form
\({\bf {\mathcal {C}}}^G_{\bar{k}}:=m_1[f_0]\max _{k<\bar{k}+1}\!\!\left( \!\frac{c^{-1}_\gamma (sk\!+\!\frac{\gamma }{2})}{ \mathcal {A}_{s}}\right)^{c^{-1}_\gamma (sk\!+\!\frac{\gamma }{2})}.\) 

The reader can observed that the positive coercive constant \(\mathcal {A}_=c_{lb}2^{\frac{\gamma }{2}}{ A_}\) , may or may not be bigger that unity, and so the power associated to the quotient \(c^{-1}_\gamma (sk+\gamma /2)/ \mathcal {A}_{s}\)  needs to be maximize between \(1<sk<s \bar{k}\) .
Therefore, recalling that \( c^{-1}_\gamma (sk\!+\!\frac{\gamma }{2})= \frac{2sk + \gamma -2}{\gamma }, \)  the corresponding constant upper \( {\bf {\mathcal {C}}}^G_{k_0}\)  bound associated to (REF ) is simply stated to be
\({\bf {\mathcal {C}}}^G_{k_0} := m_1[f_0]\max _{k\le k_0+1} \left( \!\frac{2sk+\gamma -2}{\gamma \mathcal {A}_{s}}\right)^{\!\!\frac{2sk+\gamma -2}{\gamma }}.\) 
Therefore, for any \(t\in (0, 1]\)  and moment order \(k_0\ge 0\) , the first term in that last RHS of (REF ) estimates the partial moments sum majoring with the constant \( {\bf C}^G_{k}\)  from (REF ) by \( {\bf C}^G_{sk_0}\) , and natural upper bounds for \(m^{\prime }_{sk+\frac{\gamma }{2}}\)  from (REF )
as follows
\( \nonumber \sum _{k=0}^{k_{0}} &\left( \mathcal {A}_{s} m_{s k+\frac{\gamma }{2}}[f] +m^{\prime }_{sk}[f] \right) \frac{z(t)^k}{k!} < \sum _{k=0}^{k_{0}} \left[{\mathcal {A}}_{s} m_{s k+\frac{\gamma }{2}}[f] + {\tt \bf m}_{1,sk}[f_0] \,\beta _{sk}\, R_{{\mathcal {C}}} (\delta ^{-1}_{sk}) \right.\\&\left.\qquad \qquad + \left(2\, \Vert  b\Vert _{L^1(\mathbb {S}^{d-1})} \right)^{sk+\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+1+2\gamma }\left( {\tt \bf m}_{1,sk}[f_0] \, R_{{\mathcal {C}}} (\delta ^{-1}_{sk}) +\!{\bf C}^G_{\bar{k}}(f_0, C_\Phi ,\gamma )\right) t^{-\frac{2sk\!+\!\gamma -2}{\gamma }}\right] \frac{z(t)^k}{k!}\nonumber \\&\quad < {\mathcal {A}}_{s}\left( k_0 \mathcal {R}(\delta ^{-1}_{sk_0+ \frac{\gamma }{2}} ) \max _{0<t\le 1} e^{z(t)} \right. \nonumber \\&\left. \qquad \qquad \qquad \qquad + m_1[f_0] \max _{k<k_0}\left(\left( \frac{c^{-1}_\gamma (sk+\frac{\gamma }{2})}{\mathcal {A}_{s}}\right)^{\frac{1}{c_\gamma (sk+\frac{\gamma }{2})}} \right)\sum _{k=0}^{k_0}t^{-{\frac{2sk+\gamma -2}{\gamma }}} \frac{z(t)^k}{k!}\!\right) \nonumber \\& \qquad \qquad +\, \left(2 \Vert  b\Vert _{L^1(\mathbb {S}^{d-1})}\right)^{sk_0+\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+1+\frac{\gamma }{2}}\left( {\tt \bf m}_{1,sk_0}[f_0] \mathcal {R} (\delta ^{-1}_{sk_0}) +{{\mathcal {C}}_{\bf {k}_{\text{\tiny mx}}}^G} \right)\, \sum _{k=0}^{{\bf {k}_{\text{\tiny mx}}}+1} t^{-\frac{2sk+\gamma -2}{\gamma }} \frac{z(t)^k}{k!}\nonumber \\&\quad <\left( {\mathcal {A}}_{s}k_0 \max _{0<t\le 1} e^{z(t)} +\left(2 \Vert  b\Vert _{L^1(\mathbb {S}^{d-1})}\right)^{sk_0+\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+1+\frac{\gamma }{2}}{\tt \bf m}_{1,sk_0+\frac{\gamma }{2}}[f_0] \right) \mathcal {R}(\delta ^{-1}_{sk_0+ \frac{\gamma }{2}})\nonumber \\&+ \left[ {\mathcal {A}}_{s}{\bf {\mathcal {C}}}_{k_0}^G+ (2\Vert  b\Vert _{L^1(\mathbb {S}^{d-1})})^{sk_0+\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+1+\frac{\gamma }{2}} {{\mathcal {C}}_{\bf {k}_{\text{\tiny mx}}}^G} \right]\sum _{k=0}^{k_{0}} \, {t}^{-\frac{1}{c_\gamma (sk+\frac{\gamma }{2}) }} \frac{z(t)^k}{k!}.\) 

with the constant rate factor \(\mathcal {R}(\delta ^{-1}_{sk_0+ \frac{\gamma }{2}}):=\mathcal {R}(\beta _{sk_0+ \frac{\gamma }{2}}/\mathcal {A}_{s})\) 
is from (REF ), provided that \(sk_0>1\) .
Hence, let the exponential rate \(z\)  to be
\(z(t) \le \ln e \qquad \iff \qquad e^{z(t)} \le 1,\) 

the constant factor
\({\bf {\mathcal {K}}}^G_{k_0} := \left( 1 +\left(2\Vert  b\Vert _{L^1(\mathbb {S}^{d-1})}\right)^{sk_0+\frac{{\bf {k}_{\text{\tiny mx}}}}{2}+1+\frac{\gamma }{2}} \right),\) 

and the constant \(\kappa ^G_{sk_0}\)  set
\(\kappa ^G_{sk_0}&:= \max \left\lbrace k_0 \,;\, {\tt \bf m}_{1,sk_0+\frac{\gamma }{2}}[f_0] \right\rbrace .\) 
It follows that estimate (REF ) can be expressed term of these constant factors (REF ,REF ) and (REF ), to obtain the condensed form
\(\sum _{k=0}^{k_{0}} &\left( \mathcal {A}_{s} m_{s k+\frac{\gamma }{2}}[f] +m^{\prime }_{sk}[f] \right) \frac{z(t)^k}{k!}< {\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} \mathcal {R}(\delta ^{-1}_{sk_0+ \frac{\gamma }{2}})+ {\bf {\mathcal {C}}}_{k_0}^G\sum _{k=0}^{k_{0}} \, {t}^{-\frac{2sk_0+\gamma -2}{\gamma }} \frac{z(t)^k}{k!}\right).\) 
The second term from (REF ) is estimated by the Discrete Convolution Lemma REF  set up for \(\tilde{k}=k_0\) , inequality (REF ),
now for any \(s\le \gamma /2\) ,
yields
\(2^{\frac{3\gamma }{2}-1}C_{\Phi }\mu _{sk_0}\sum _{k=k_0+1}^n \frac{z^k(t)}{k!} S_{sk}[f] \le 2^{\frac{3\gamma }{2}-1}C_{\Phi } \mu _{sk}\ E^n _{s} (z(t),t)\, I^n _{s,\gamma }(z(t),t),\quad \text{for any} \ 0< t \le 1. \) 
Then, the last term from the right hand side of (REF ) is gathered by the fact that \(I^{n-1}_{s,\gamma }(z(t),t)\le I^{n}_{s,\gamma }(z(t),t)\)  together with (REF ) and (REF ) to obtain upper bounds
to the ODI for \( E^{n}_{s}(t,z\,t)\)  as follows
\(\frac{d}{dt} E^{n}_{s}(t,z(t) ) &\le {\bf {\mathcal {K}}}^G_{k_0}\left(\max _{0<t\le 1} e^{z(t)} \, \kappa ^G_{sk_0}\mathcal {R}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}})+ \mathcal {A}_{s} {\bf {\mathcal {C}}}^G_{k_0}\sum _{k=0}^{k_{0}} \, {t}^{-\frac{1}{c_\gamma (sk)}} \frac{z(t)^k}{k!}\right)\nonumber \\&\qquad \quad - I^n _{s,\gamma }(z(t),t)\left( \mathcal {A}_{s} - z^{\prime }(t) - 2^{\frac{3\gamma }{2}-1}C_{\Phi } \mu _{sk_0}\ E^n _{s} (z(t),t) \right)\quad \text{for any} \ \ 0<t\le 1,\) 
Further, the negative term proportional to the shifted moments partial sums \(I^n _{s,\gamma } (z(t),t)\) , is estimated from below by
\({I}^n_{s; \gamma }(z(t),t) &:= \sum _{k=0}^{n} \, \frac{z^k(t)}{k!} m_{sk+\gamma /2}(t)=\frac{1}{z(t)} \left( \sum _{k=0}^{n}(k+1) \, \frac{z^{k+1}(t)}{(k+1)!} m_{sk+\gamma /2}(t) \right) \\&\ge \frac{1}{z(t)} \left( \sum _{k=1}^{n+1} \, \frac{z^k(t)}{k!} m_{s k}(t) \right) > \frac{1}{z (t)} \left(E^{n} _{s} (z(t),t) - m_0[f_0]\right)\ge 0, \nonumber \) 

which implies, under conditions \(0<t\le 1\)  and \(0< s\le \gamma /2\)
\(\frac{d}{dt} E^{n}_{s}(t,z(t) ) &<{\bf {\mathcal {K}}}^G_{k_0}\left(\max _{0<t\le 1} e^{z(t)} \, \kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) +\mathcal {A}_{s} {\bf {\mathcal {C}}}^G_{k_0}\sum _{k=0}^{k_{0}} \, {t}^{-\frac{1}{c(sk_0+\frac{\gamma }{2}) }} \frac{z(t)^k}{k!}\right)\nonumber \\&\qquad \quad \frac{(E^{n} _{s} (z(t),t) - m_0[f_0])}{z(t)} \left(\mathcal {A}_{s} - z^{\prime }(t) - 2^{\frac{3\gamma }{2}-1}C_{\Phi } \mu _{sk_0}\ E^n _{s} (z(t),t) \right).\) 
Since the latter term in the right side needs to be an absorption term, we require the condition on the exponential rate \(z(t)\)  that maximizes is the largest possible one satisfying the condition
\(z^{\prime }(t) \le \frac{\mathcal {A}_{s}}{2} \ \ \ \ \text{for any}\ 0 < t <1.\) 

In particular, after performing a time integration recalling that \(z(0)=0\) , the natural choice is \(0\le z(t) := zt \le \frac{\mathcal {A}_{s}}{2} t\) , and therefore the \(z\)  factor must satisfy
\(z \le \frac{\mathcal {A}_{s}}{2}\) 
Hence,
\(\frac{d}{dt} E^{n}_{s}(t,z(t) ) &< {\bf {\mathcal {K}}}^G_{k_0} \left(\max _{0<t\le 1} e^{z(t)} \,\kappa ^g_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}})+\mathcal {A}_{s} {\bf {\mathcal {C}}}^G_{k_0} \sum _{k=0}^{k_{0}} \, {t}^{-\frac{1}{c_\gamma (sk)}} \frac{z(t)^k}{k!}\right)\nonumber \\&\qquad \qquad - \frac{(E^{n} _{s} (z(t),t) - m_0[f_0])}{z(t)} \left( \frac{\mathcal {A}_{s}}{2} - 2^{\frac{3\gamma }{2}-1}C_{\Phi } \mu _{sk_0}\ E^n _{s} (z(t),t)\right).\) 
Now, invoking (REF ) for the choice \(z(t)=\frac{\mathcal {A}_}{2} t\) , then for any \(t\in (0,1]\) , it follows
\(\sum _{k=0}^{k_{0}} & \, {t}^{-\frac{1}{c_\gamma (sk)}} \frac{z(t)^k}{k!}= \sum _{k=0}^{k_{0}} t^{-\frac{2sk -\gamma + 2}{\gamma }} t^k \frac{1}{k!}\left( \frac{\mathcal {A}_}{2}\right)^k = \sum _{k=0}^{k_{0}} t^{ {k} (1-\frac{2s}{\gamma }) +(\frac{2}{\gamma }-1)} \frac{1}{k!} \left( \frac{\mathcal {A}_}{2}\right)^k <e^{\frac{\mathcal {A}_{s}}{2} } , \) 

for any \(s\in (0,\gamma /2]\)  and \( \gamma \in (0,2]\) .
To this end, a reduced ODI for the partial sums of \(sk^{th}\) -moments arises
\(\frac{d}{dt} E^{n}_{s}(t,zt ) &< e^{\frac{A_{s}}{2} } {\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right) \\& \qquad - \frac{(E^{n} _{s} (t,zt) - m_0[f_0])}{zt} \left( \frac{\mathcal {A}_{s}}{2} - 2^{\frac{3\gamma }{2}-1}C_{\Phi } \mu _{sk_0}\ E^n _{s} (z(t),t)\right) .\nonumber \) 
Next, recall that
\(m_0[f_0] < \int _{\mathbb {R}^d} f_0(v)\langle v\rangle ^{2k_{in}} dv =: M_G,\quad \text{for} \ \ n\in \mathbb {N}\,.\) 
Therefore, since \(m_0[f_0](0)=E^{n}_{s}(0 , z(0)) <M_G\) , proceeding in an analog way that we did for the exponential propagation, we set the following sequence of times \(t_n\) , motivated by the search of the exponential rate \(z\)  small enough that enables an upper bound for \(E^{n}_{s}(t,zt)\) , given by
\(t_n:=\sup \big \lbrace  t\in (0,1]\,:\, E^{n}_{s}(\tau , z(\tau )) \le M_G, \; \forall \, \tau \in [0,t] \big \rbrace \,,\quad \text{for}\ \ \ n\in \mathbb {N},\) 
where \(t_n\le 1\)  is the first touching time where \(E^{n}_{s}(t , z(t)) = M_G\) , with
\(M_G\)  is the initial moment with \(k_{in}^{th}\) -order Lebesgue weighted tail.
In addition invoking moment generation estimates up to \(k=n\) , it holds that
\(t^{k} \, m_{sk} (t) \le {\bf {\mathcal {C}}}^G_{sk}\, t^{k(1-\frac{2}{\gamma }s)+\frac{2}{\gamma }}\,,\quad \text{for}\quad 1\le k \le n\,,\) 
so that, if \(s\in (0,\frac{\gamma }{2}]\)  and \(\gamma \in (0,2]\) , the limit as \(t\searrow 0\)  of this product vanishes. Hence,
\(E^{n}_{s}(t , z(t))&:= \sum _{k=0}^{n} \, \frac{(zt)^k}{k!} m_{sk}(t)= m_0[f_0]+\sum _{k=1}^{n} \, \frac{(zt)^k}{k!} m_{sk}(t) \le m_0[f_0] \\& \qquad \qquad + \left(e^{\frac{\mathcal {A}_{s}}{2}} -1\right) {\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right)\, t^{\frac{2}{\gamma }}\,.\) 

Consequently, since \(m_0<M_G\) , it holds that
\(E^{n}_{s}(t , z(t)) \le M_G \quad \text{in the interval} \quad 0 < t\le \left(\frac{M_{G}-m_0[f_0]}{ \left(e^{\frac{\mathcal {A}_{s}}{2}} -1\right)\, {\bf {\mathcal {K}}}^G_{k_0} \, \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right) }\right)^{\frac{\gamma }{2}}\,.\) 
This guarantees that the differential estimate (REF ) satisfied by \(E^{n}_{s}(t , z(t))\)  holds in the non-empty interval \(t\in (0,t_n]\)  where
\(1\ge t_{n}\ge \left(\frac{M_{G}-m_0[f_0]}{ \left(e^{\frac{\mathcal {A}_{s}}{2}} -1\right) {\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right)}\right)^{\frac{\gamma }{2}}\,, \quad \text{for all}\quad n\in \mathbb {N}\,,\) 
or equivalently, the set (REF ) is non-empty.
In addition, due to the fact that the contractive factor \(0<\mu _{sk_0} \searrow 0\) , with a sufficiently large \(k_0:=k_0(,M_G)\) , renders the lower bound estimate
\(&\frac{\mathcal {A}_{s}}{2} - 2^{\frac{3\gamma }{2}-1}C_{\Phi } \mu _{sk_0}\ E^n _{s} (t,zt) \ge \frac{1}{2} \mathcal {A}_{s} - 2^{\frac{3\gamma }{2}-1}\,C_{\Phi }\, \mu _{sk_0}\, M_G \ge \frac{ \mathcal {A}_{s}}{4},\) 

to hold for the precise, uniform in \(n\in \mathbb {N}\) , choice of
\(&\qquad \qquad \qquad \qquad \qquad \mu _{s (k_{0}-1)} \le \frac{\mathcal {A}_{s} }{2^{\frac{3\gamma +2}{2}} C_{\Phi } M_G}, \qquad \text{for any}\quad 0<s \le \frac{\gamma }{2} \quad \text{and} \quad 0< t\le t_n\le 1.\) 

Recalling Remark REF  that stated if the angular transition \(b(\hat{u}\cdot \sigma ) \in L^p(S^{d-1})\) ,  for \(1<p\le \infty \) , then invoking (REF ) or (REF ), the \(k_0\)  moment becomes explicit
\(s k_0> \mathcal {O}\left(\frac{2^{\frac{3\gamma +4}{2}} C_{\Phi } M_G}{ \mathcal {A}_{s}}\right)^{\frac{p}{p-1}}, \qquad \text{for} \ k_0>\ \text{and}\ 0<s\le 1.\) 

Therefore, directly from estimate (REF ), substituting \(z(t)=zt\)  it follows that
\( \frac{d}{dt}E^{n}_{s}(t,zt) &< 2e^{\frac{\mathcal {A}_{s}}{2}}{\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right) -\frac{\mathcal {A}_{}}{4zt} \big (E^{n} _{s} (t,zt) - m_0[f_0] \big )\\&\text{valid for}\quad 0 < t\le t_n\le 1 \quad \text{and} \quad s\in \left(0, \frac{\gamma }{2}\right]\, .\nonumber \) 

where the coefficients of this new ODI are uniform in \(n\) .
Hence, in order to find bound to solutions of the initial value problem to inequality (REF ), it is convenient to set \(X^{n}_{k_0}(t,zt):= E_{s}^n(t,zt) - m_0[f_0]\ge 0\) , to obtain the contractive ODI
\( &\frac{d}{dt}X^n(t,zt) < 2e^{\frac{\mathcal {A}_{s}}{2}} \, {\bf {\mathcal {K}}}^G_{k_0} \,\left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right)- \frac{\mathcal {A}_{s}}{4zt}\,X^{n} (t,zt)\,,\qquad 0<t\le t_n\le 1\,, \nonumber \\&X^n(0,0):= \lim _{t\rightarrow 0^+} E_{s}^n(t,zt) - m_0[f_0]=0 \qquad \text{its inital data}.\) 
A function satisfying the differential inequality (REF ) is controlled by direct integration of the inequality \(\bar{X}^{\prime }+c\, t^{-1} \bar{X} < K\)  after using the multiplicative factor \(t^c\)  so that \((t^c \bar{X})^{\prime } \le t^c\,K\) , for \(t>0\) . We arrive to
\(\bar{X} < \bar{X}(0) + t(1+c)^{-1} B\,,\quad t>0\,.\) 
When this super solution is applied with \(c:= \frac{\mathcal {A}_{}}{2z}\)  and \(K:= 2e^{{\mathcal {A}_{}}/2}C^g_{\gamma (k_0+1)/2} ( 1+\mathcal {A}_{s})\) , and initial \(\bar{X}(0)=0\)  the target function \(X^n(t,zt)\)  is then controlled by
\( X^n(t,zt) &< \frac{4zt}{\mathcal {A}_{s} +4z} 2e^{\frac{\mathcal {A}_{s}}{2}}\, {\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right) \\&\ \ \ < \frac{8zt}{\mathcal {A}_{}} e^{\frac{\mathcal {A}_{s}}{2}} \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right) \qquad \text{for any}\ 0<t\le t_n\le 1. \nonumber \) 

Therefore choosing \(zt>0\)  such that
\( &\frac{8zt}{\mathcal {A}_{s}} e^{\frac{\mathcal {A}_{s}}{2}} {\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right) < M_G- m_0[f_0], \nonumber \\\text{or equivalently, }&\ \ \ \ \ \ \ \\&{0<zt} < \frac{(M_G-m_0[f_0]) \mathcal {A}_{s} e^{-\frac{\mathcal {A}_{s}}{2}} }{8 \, {\bf {\mathcal {K}}}^G_{k_0}\left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right) }, \qquad \text{for any}\ 0<t\le t_n\le 1,\nonumber \) 

implies that
\(E_{s}^n(t,zt) - m_0[f_0] &< \frac{8zt}{\mathcal {A}_{s}} e^{\frac{\mathcal {A}_{}}{2}}{\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} {\mathcal {R}}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}}) + {\bf {\mathcal {C}}}^G_{k_0} \right) < M_G -m_0[f_0]\qquad \text{for any}\ 0<t\le t_n\le 1,\) 

that is,
\(E_{s}^n(t,zt) < M_G, \qquad \text{for any}\ 0<t\le t_n\le 1.\) 

Since \(t\le 1\)  we conclude that any constant parameter \(z\) , uniformly in \(n\in \mathbb {N}\) , satisfying
\(0<z< \min \left\lbrace  \frac{\mathcal {A}_{s}}{2}\ ;\ \mathcal {A}_{s}\frac{ \mathcal {R}(\delta _{sk_0+\frac{\gamma }{2}}) (M_G-m_0[f_0]) \, e^{-\frac{\mathcal {A}_{s}}{2}} }{8\, {\bf {\mathcal {K}}}^G_{k_0} \left(\kappa ^G_{sk_0} + {\bf {\mathcal {C}}}^G_{k_0} \, {\mathcal {R}}(\delta _{sk_0+\frac{\gamma }{2}}) \right) } \right\rbrace  ,\) 

is the admissible rate, such that \(E^{n}_{s}(t,zt)<M_G\)  for \(0<t\le t_n\le 1\) , for any \(s\in (0,\gamma /2]\)  and the constant \( \kappa ^g_{sk_0}\)  defined in (REF ).
Since the rate factor \(1<{\mathcal {R}}(\beta _{sk_0+\frac{\gamma }{2}}/\mathcal {A}_{s})=\mathcal {R}(\delta ^{-1}_{sk_0+\frac{\gamma }{2}})\)  and finite, then the rate \(z\)  will not degenerate for any \(0<\gamma \le 2\) . In addition, recalling \(\mathcal {A}_{} := c_{lb} 2^{\frac{\gamma }{2}} A_{s}\) , the form of the rate of decay for the exponential tail, depending on the coercive factor \(A_{s}\) , by replacing on (REF ), to obtain the rate (REF ) in the statement of generation of exponential moments.
Note that the inequality (REF ) for the choice of such rate \(z(t)=zt\)  is strict, that means there is no touching time in \((0,t_n]\) , so it must be the case that \(t_n=1\)  due to time continuity of the moments evolution.
Sending \(n\rightarrow \infty \)  one concludes
\( E_{s}(t,zt) < M_G, \quad \ \text{for all}\quad 0\le t\le 1,\quad s\in \left(0, \frac{\gamma }{2}\right]\,.\) 
Note that the dependence of \(z\)  is in terms of the conserved quantities, that is, the mass and energy of \(f\)  and on \(M^\star _G\) , defined as \(M^\star _G:=\sup _{t\ge 0} m_{k_{in}}[f(t)]\) , replacing \(M_G\) . All such quantities are independently on time. Therefore, by the time-invariance of the equation, one redoes the argument in the intervals \(t\in [{\bf {k}_{\text{\tiny mx}}}/2,1+{\bf {k}_{\text{\tiny mx}}}/2]\) , with \({\bf {k}_{\text{\tiny mx}}}=1,2,\cdots \) , to conclude that
\( E_{s}(t,zt) \le M^*_G, \quad \ \text{for all}\quad t\ge 0,\quad s\in \left(0, \frac{\gamma }{2}\right]\,.\) 
The proof Theorem REF  is now completed.

Convolution inequalities and gain of integrability for the collision operator
After the groundbreaking work of Arkeryd [2]} in 1982 on propagation of \(L^\infty \)  estimates for bounded angular cross-sections, in late eighties and nineties Gustaffson [3]}, Wennberg [4]}, Lions [5]}, among others, studied \(L^r({\mathbb {R}}^d)\)  estimates, for \(1\le r <\infty \) , for the collision operator. Lions introduced the concept of gain of integrabilty, by meaning that it is possible to obtain smoothing properties of the collision operator under certain regularity conditions of the potential kernel. His work culminated in estimates of the form
\(\int _{\mathbb {R}^{d}} Q^+(f, g)(v,t)f(v,t) dv \le C(\Vert g\Vert _{L^1}) \Vert  f\Vert _{L^2}^{2-\delta },\quad \text{for some} \quad \delta \in (0,1)\,.\) 
Yet, both constants \(C(\Vert g\Vert _{L^1})\)  and \(\delta \in (0,1)\)  where shown to exist but hard to calculate. These results provided a venue to obtain energy estimates suitable for the study of the evolution of \(L^r\) -norms for solutions to the Boltzmann equation later in [6]}, [7]}. Such initial results where limited to potentials satisfying certain regularity conditions in both kinetic and angular variables, including angular cut-off near \(\hat{u}\cdot \sigma =\pm 1\) .
The new approach presented here comprises recent versions of functional inequalities that will enable to address the study of the propagation for the \(L^{r}\) -norms for the Boltzmann equation, not only under general integrable angular kernels for the first time, but also following estimates based in previous work [8]}, [9]}, [10]}, it is possible to obtain explicit constants important to describe the dependence of the key parameters at play such as the angular transition function.
Motivated by this goal, we revisit the Young's inequality's proof with techniques introduced by the authors and Carneiro in [8]}, [9]}, based on radial symmetrization techniques applied to the weak form of the gain collision Boltzmann operator, not by invoking the classical representation (REF , REF ), but rather written in terms of center of mass and relative velocity coordinates as described in (REF ) taking the form \(v^{\prime }_* -v= -u^{+}:=-\tfrac{1}{2}\big (u+|u|\sigma \big )\)  and \(v^{\prime }-v = u^{-}:=\tfrac{1}{2}(-u+|u|\sigma )\) , as follows.
Young's type inequality for the gain collision operator
In the late eighties, it was show in [3]} that the gain part of the collision operator satisfies Young's convolution inequalities. Indeed, as the reader may have noticed, the gain collision operator is a special type of nonlinear convolution between the entries and the collision kernel. The reader can find in [8]}, [9]} the following version of such convolution inequalities. These estimates will be important for the propagation of \(L^r\) -norms, for the range \(1< r \le \infty \) , to be shown as an application to the Boltzmann equation in the next section.
Theorem 28 (Young's type inequality) 
Let \(1 \le p,\,q,\,r \le \infty \)  with \(1/p + 1/q = 1 + 1/r\) . Assume that
\(B(|u|,\hat{u}\cdot \omega )=b(\hat{u}\cdot \omega )\in L^1(\mathbb {S}^{d-1})\,.\) 
Then, the bilinear operator \(Q^{+}\)  satisfies
\(\left\Vert  Q^{+}(f,g)\right\Vert _{L^{r}({\mathbb {R}}^d)} \le C_{b}(p,q) \,\Vert f\Vert _{L^p({\mathbb {R}}^d)} \, \Vert g\Vert _{L^q({\mathbb {R}}^d)}.\) 
The constant \(C_{b}(p,q)\)  is given by
\(\begin{split}C_{b}(p,q)=&\left(\int _{\mathbb {S}^{d-1}}\left(\tfrac{1 + e_{1}\cdot \sigma }{2}\right)^{-\tfrac{d}{2r^{\prime }}}b(e_{1}\cdot \sigma )d\sigma \right)^{\tfrac{r^{\prime }}{p^{\prime }}}\left(\int _{\mathbb {S}^{d-1}}\left(\tfrac{1 - e_{1}\cdot \sigma }{2}\right)^{-\tfrac{d}{2r^{\prime }}}b(e_{1}\cdot \sigma )d\sigma \right)^{\tfrac{r^{\prime }}{q^{\prime }}}.\end{split}\) 

In the case \((p,q,r) = (1,1,1)\)  the constant \(C_{b}(1,1)\)  is understood as
\(C_{b}(1,1) = \int _{\mathbb {S}^{d-1}}b(e_{1}\cdot \sigma )d\sigma = \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}.\) 
In the cases that \(p=1\)  or \(q=1\)  one interprets \((\cdot )^{r^{\prime }/p^{\prime }}=(\cdot )^{0} = 1 \)  and \((\cdot )^{r^{\prime }/q^{\prime }}=(\cdot )^{0} = 1\)  respectively.

It is of interest to recall that Young's inequality framework's proof that developed by introducing
a weighted \(\mathcal {P}\) -operator as a function of the the angular integration, written in center of mass and relative velocity coordinates (REF ) is defined by
\(\mathcal {P}({\it \Psi }, {\it \Phi })(u) &:= \int _{\mathbb {S}^{d-1}} {\it \Psi }(u^-(u,\sigma ) {\it \Phi }(u^+(u,\sigma )) b(\hat{u} \cdot \sigma ) d\sigma ,\ \qquad u^{\pm }:=\tfrac{1}{2}\big (u\pm |u|\sigma \big )\,\) 
Indeed, the operator \(\mathcal {P}\)  in (REF ) used in the weak formulation plays the central role of angular averaging occurring in \(Q^{+}(f,f)(v)\)  as will be explained below, after Theorem REF .
From this representation, it is clear that the integrability properties of \(\mathcal {P}\)  are central to the understanding of the integrability properties of \(Q^{+}\)  itself.
More precisely, using radial symmetrization one can readily prove the following result,
Theorem 29 
Let \(1 \le p,q,r \le \infty \)  with \(1/p+1/q=1/r\) . Then,
\(&\Vert \mathcal {P}_b({\it \Psi }, {\it \Phi })\Vert _{L^r(\mathbb {R}^{d})}\le \tilde{C}_{b}(p,q) \Vert {\it \Psi }\Vert _{L^p(\mathbb {R}^{d})}\Vert \it \Phi \Vert _{L^q(\mathbb {R}^{d})}, \nonumber \\\text{where}\qquad \qquad \qquad &\ \\&\tilde{C}_{b}(p,q)=\int _{\mathbb {S}^{d-1}} \left(\tfrac{1 - e_{1}\cdot \sigma }{2}\right)^{-\frac{d}{2p^{\prime }}} \left(\tfrac{1 + e_{1}\cdot \sigma }{2}\right)^{-\frac{d}{2q^{\prime }}}\, b\big ( e_{1}\cdot \sigma \big )d\sigma . \nonumber \) 

In addition, if either functions \(\it \Psi \)  or \(\it \Phi \)  are unity, but not simultaneously, then the statement (REF ) is modified by setting \(q=\infty ,\)  \(r=p\) , since \(1=\phi \in L^\infty (\mathbb {R}^{d}\times \mathbb {S}^{d-1})\)  to estimate, for  \(\mathbb {S}_{\mp }^{d-1}:= \lbrace e_{1}\cdot \sigma \lessgtr 0 \rbrace , \)
\(\Vert \mathcal {P}^-_b({\it \Psi },1)\Vert _{L^p(\mathbb {R}^{d})}:=\Vert \mathcal {P}_b({\it \Psi },1)\Vert _{L^p(\mathbb {R}^{d})}&\le \tilde{C}_{b}(p,\infty ) \Vert \it \Psi \Vert _{L^p(\mathbb {R}^{d})},\nonumber \\[4pt]\ \text{with}\ \tilde{C}_{b}(p, \infty )&=\int _{\mathbb {S}_-^{d-1}} \left(\tfrac{1 -e_{1}\cdot \sigma }{2}\right)^{-\frac{d}{2p^{\prime }}} b\big ( e_{1}\cdot \sigma \big )d\sigma , \nonumber \\[4pt]\text{or setting}\ p=\infty , r=q,\ \text{for} \ 1=\psi \in L^\infty (\mathbb {R}^{d}\times \mathbb {S}^{d-1}), &\ \text{to estimate} \qquad \qquad \qquad &\ \\[4pt]\Vert \mathcal {P}^+_b(1,{\it \Phi })\Vert _{L^q(\mathbb {R}^{d})}:= \Vert \mathcal {P}_b(1,{\it \Phi })\Vert _{L^q(\mathbb {R}^{d})}&\le \tilde{C}_{b}(\infty ,q) \Vert \it \Phi \Vert _{L^q(\mathbb {R}^{d})}, \nonumber \\[4pt]\text{with} \ \ \tilde{C}_{b}(\infty ,q)&=\int _{\mathbb {S}_+^{d-1}} \left(\tfrac{1 + e_{1}\cdot \sigma }{2}\right)^{-\frac{d}{2q^{\prime }}} b\big ( e_{1}\cdot \sigma \big )d\sigma . \nonumber \) 

All details on how to proof Therorem REF  can be found at [8]}, as well as at [9]} for further applications to Young's inequalities for gain operators.
Theorem REF  is the base to prove Theorem REF . It also explains the peculiar expression of the constants in such theorem. Interestingly, the constants given in Theorem REF  are known to be sharp in the sense of functional inequalities.
However, it is important to notice that the definition of \(\mathcal {P}_b(\psi ,\phi )(u)\)  in (REF ) depends on test functions \(\psi \)  and \(\phi \)  acting on pairs \(u^\pm (u,\sigma )= \frac{1}{2}(|u|\sigma \pm u)\) , respectively. In addition,
\(|u^\pm | = |u| \,\frac{1}{2}(1\pm e_{1}\cdot \sigma )^{1/2}\) . This last identity means that \( |u^+|=|u| \,\frac{1}{2}(1 + e_{1}\cdot \sigma )^{1/2}>0 \) , i.e. non-vanishing, whenever \(\sigma \in \mathbb {S}_+^{d-1}:= \lbrace e_{1}\cdot \sigma >0\rbrace \)  and, analogously, \( |u^-|=|u| \,\frac{1}{2}(1 - e_{1}\cdot \sigma )^{1/2}>0 \)  whenever \(\sigma \in \mathbb {S}_-^{d-1}:= \lbrace e_{1}\cdot \sigma <0\rbrace .\)
Thus, while the reader can observe that the definition of the constant \(C_{b}(p,q)\)  in (REF ) may contain two singularities at the points \(e_{1}\cdot \sigma =\pm 1\) , (i.e in the north and south poles of the scattering sphere where \(e_1=\hat{u}\)  fixed),
it becomes transparent in the next paragraph that the weak formulation associated to the bilinear collisional form \(Q^+(f,g)(v)\) , written in center of mass and relative velocity or scattering direction coordinates (REF ), takes a form in terms of the simpler estimates for \(\mathcal {P}\)  from (REF ) described in (REF ), by associating the definition of the domain of spherical integration of the test functions to whether it is evaluated on \(u^+(u,\sigma )\)  with \(\sigma \in \mathbb {S}_+^{d-1}\) ; or \(u^-(u,\sigma )\)  with \(\sigma \in \mathbb {S}_-^{d-1}\) .
Indeed, the characterization of the operator \(\mathcal {P}({\it \Psi }, {\it \Phi })(u):= \int _{\mathbb {S}^{d-1}} {\it \Psi }(u^+(u,\sigma ) {\it \Phi }(u^-(u,\sigma )) b(\hat{u} \cdot \sigma ) d\sigma ,\)  in the weak formulation of the \(Q^+(f,g)(v)\)  is conveniently written in scattering direction coordinates (REF ), namely \(v^{\prime }=v+u^-(\sigma )\)  and \(v^{\prime }_*=v-u^+(\sigma )\) .
Thus, recalling that \(\tau \)  is the translation operator \(\tau _v\psi (x) := \psi (x-v)\)  and \(\mathcal {R}\)  the reflexion operator \(\mathcal {R}\psi (x):=\psi (-x)\) , then for any given test function \(\psi \)  set \({\it \Psi }(u^-(u,\sigma )):= \psi (v+u^-(u,\sigma ))= (\tau _{-v} \psi )(u^-(u,\sigma ))\) , and
\({\it \Phi }(u^+(u,\sigma )) := \psi (v-u^+(u, \sigma ))= (\tau _{-v}\mathcal {R} \psi )(u^+(u,\sigma ))\) 
and applied the estimates from Theorem  REF
\(\mathcal {P}({\it \Psi }, {\it \Phi })(u)&:= \int _{\mathbb {S}^{d-1}} {\it \Psi }(u^-(u,\sigma ))\,{\it \Phi }(u^+(u,\sigma )) b(\hat{u} \cdot \sigma ) d\sigma ,\) 
Thus, splitting the spherical integration domain into \(\sigma \in \mathbb {S}^{d-1} = \lbrace \hat{u}\cdot \sigma <0 \rbrace \cup \lbrace \hat{u}\cdot \sigma \ge 0 \rbrace  =: \mathbb {S}_-^{d-1} \cup \mathbb {S}_+^{d-1}\) 
is given by, the weak form of \(Q^+(f,g)(v)\)  becomes
\(\int _{\mathbb {R}^d} \!Q^{+}&(g,f)(v)\psi (v) dv = \int _{\mathbb {R}^{d}} f(v)\int _{\mathbb {R}^{d}}\Phi (|u|) g(v-u)\left(\int _{\mathbb {S}_-^{d-1}}\psi (v+u^-(\sigma )) {b(\hat{u} \cdot \sigma )} d\sigma \right)\!(u) du dv \nonumber \\&+ \int _{\mathbb {R}^{d}} g(v_*)\int _{\mathbb {R}^{d}}\Phi (|u|) f(v_*+u)\left(\int _{\mathbb {S}_+^{d-1}}\psi (v-u^+(\sigma )) {b(\hat{u} \cdot \sigma )} d\sigma \right)\! (u)du dv_*\nonumber \\&=:\!\int _{\mathbb {R}^{d}}\! f(v)\int _{\mathbb {R}^{d}}\Phi (|u|) ,\tau _{-v}\mathcal {R}g(u) \left[ \mathcal {P}^-_{b}\left((\tau _{-v}\psi )(u^-), 1\right)\!+\! \mathcal {P}^+_{b}(1,(\tau _{-v}\mathcal {R}\psi )(u^+) \right]\!(u) du dv\nonumber \\&=:\int _{\mathbb {R}^{d}}\! f(v)\int _{\mathbb {R}^{d}}\Phi (|u|) \, \tau _{-v}\mathcal {R}g(u)\, \left[ \mathcal {P}^-_{b}\left({\it \Psi }(u^-), 1\right)\! + \! \mathcal {P}^+_{b}(1,{\it \Phi })(u^+) \right]\!(u) du dv,\) 

since \(\it \Psi (u^-) := (\tau _{-v}\psi )(u^-)\)  and \(\it \Phi (u^+) := (\tau _{-v}\mathcal {R}\psi )(u^+)\) .
The potential part of the transition scattering rates
\( \Phi (|u|)\)  is the potential function defined on (REF ) under the assumption (REF ), where for the time being, we illustrate the technique for the case of \(\gamma =0\) , which implies \(0<c_{\Phi }\le \Phi (|u|)\le C_{\Phi }<\infty \)  uniformly in \(|u|\) . Section 8.2 will extended the following estimate to general potential, satisfying this conditions, for any \(\gamma \in (0,2]\) .
Therefore, an estimate of (REF ) needs to evaluate the first term noting that the test function \(\psi (v+u^-(\sigma ))\)  acts on the first component of the \(\mathcal {P}\)  operator (REF ), while the second term now the test function, not only is evaluated at a different velocity vector, \(\psi (v-u^+(\sigma ))\) , but also acts on the second component of the \(\mathcal {P}\) .
This observation implies the constant \(\tilde{C}_{b}(p,q)\)  can be calculated finite using the estimates (REF ) from Theorem REF .
Therefore, since the first term of identity (REF ) has test function \(\psi (v-u^-(u,\sigma ))\)  acting on the first component of the \(\mathcal {P}\)  from (REF ) while the test function in the second component \(\phi \)  of the \(\mathcal {P}\)  operator is taken to be unity,
hence, Theorem REF  can be applied with estimate
(REF ) by choosing \(r = p = s\)  and \(q=\infty \) ,
to obtain, after invoking Hölder's inequality, finite inequality
\(\int _{\mathbb {R}^{d}}(\tau _{-v} \mathcal {R}g)(u)\mathcal {P}^-_{b}({\it \Psi }, 1)(u) du\le \tilde{C}_{b}(s,\infty )\Vert \tau _{-v}g\Vert _{L^{s}}\Vert  \tau _{-v}\psi \Vert _{L^{s^{\prime }}}=\tilde{C}_{b}(s,\infty )\Vert g\Vert _{L^{s}}\Vert  \psi \Vert _{L^{s^{\prime }}}\,,\) 
with
\(\tilde{C}_{b}(s,\infty )=\int _{\mathbb {S}_-^{d-1}} \left(\tfrac{1 - e_{1}\cdot \sigma }{2}\right)^{-\frac{d}{2s^{\prime }}} b( e_{1}\cdot \sigma )d\sigma \le 2^{\frac{d}{2s^{\prime }}} \Vert b\Vert _{L^{1}(\mathbb {S}_-^{d-1})}\,.\) 
Analogously, the second term in (REF ) has the first component of \(\mathcal {P}\)  from (REF ) in \(L^\infty (\mathbb {R}^{d}\times \mathbb {S}_+^{d-1})\) , while the test function \(\psi (v-u^+(\sigma ))\)  is acting on the second component of the \(\mathcal {P}\)  on the integrating domain \( \mathbb {S}_-^{d-1}\) . Therefore, Theorem REF  is applied with estimate (REF ), with the choice \(q=r=s\)  and \(p=\infty \) , yielding
\(\int _{\mathbb {R}^{d}}(\tau _{-v}g) (u)\mathcal {P}^+_{b}(1,{\it \Phi })(u) du \le \tilde{C}_{b}(\infty ,s)\Vert  \tau _{-v}g \Vert _{L^{s}}\Vert  \tau _{-v}\mathcal {R}\psi \Vert _{L^{s^{\prime }}}=\tilde{C}_{b}(\infty ,s)\Vert  g \Vert _{L^{s}}\Vert  \psi \Vert _{L^{s^{\prime }}(\mathbb {S}_+^{d-1})}\,,\) 

where
\(\tilde{C}_{b}(\infty ,s)=\int _{\mathbb {S}_+^{d-1}} \left(\tfrac{1 + e_{1}\cdot \sigma }{2}\right)^{-\frac{d}{2s^{\prime }}} b( e_{1}\cdot \sigma )d\sigma \le 2^{\frac{d}{2s^{\prime }}} \Vert b\Vert _{L^{1}(\mathbb {S}_+^{d-1})}\,.\) 
Combining these two last estimates, it follows that
\(\int _{\mathbb {R}^d}Q^{+}_{b}(fig)(v) \psi (v) dv &\le \tilde{C}_{b}(s,\infty )\Vert  g\Vert _{L^1} \Vert f\Vert _{L^{s}}\Vert  \psi \Vert _{L^{s^{\prime }}} + \tilde{C}_{b}(\infty ,s)\Vert f\Vert _{L^1} \Vert  g \Vert _{L^{s}}\Vert  \psi \Vert _{L^{s^{\prime }}}\\&\le 2^{\frac{d}{2s}}\Vert  \psi \Vert _{L^{s^{\prime }}} \Big (\Vert b\Vert _{L^{1}(\mathbb {S}_-^{d-1})}\Vert  g\Vert _{L^1} \Vert f\Vert _{L^{s}} + \Vert b\Vert _{L^{1}(\mathbb {S}_+^{d-1})} \Vert f\Vert _{L^1} \Vert  g \Vert _{L^{s}} \Big )\,, \nonumber \) 

or, by duality,
\(\Vert Q^{+}_{b}(f,g) \Vert _{L^{s}(\mathbb {R}^d)} \le 2^{\frac{d}{2s}}\Big (\Vert b\Vert _{L^{1}(\mathbb {S}_-^{d-1})}\Vert  g\Vert _{L^1} \Vert f\Vert _{L^{s}} + \Vert b\Vert _{L^{1}(\mathbb {S}_+^{d-1})}\Vert f\Vert _{L^1} \Vert  g \Vert _{L^{s}} \Big )\,.\) 
In the quadratic case \(f=g\)  estimate (REF ) reduces to
\(\Vert Q^{+}_{b}(f,f) \Vert _{L^{s}(\mathbb {R}^d)} \le 2^{\frac{d}{2s}}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f\Vert _{L^1} \Vert f\Vert _{L^{s}}\,,\qquad s\in [1,\infty ]\,.\) 
The expression (REF ) does not necessarily holds for gas mixtures with disparate masses or inelastic interacting systems since the definition of vectors \(u^\pm :=u^\pm (u, \sigma )\)  do not enjoy the symmetry \(u^{\pm }(u,-\sigma )=u^{\mp }(u, \sigma )\)  in such cases. Such vectors will change depending on the masses or restitution coefficients respectively. This, of course, makes the analysis a bit more subtle.

The Carleman representation, mixing convolution, and gain of integrability
Carleman in his work [18]} represents the gain part of the collisional integral with the integration in the sphere replaced by an integration along special planes. In a sense, he introduced a trick to project the sphere in \(\mathbb {R}^{d-1}\)  in the context of the collision operator. A similar trick has been implemented in the context of radiative transfer in the non-cutoff case in [19]} and has the benefit of flatten the geometry of the operator at the expense of adding weights in the integration. This representation can be quite handy in the study of fine properties of the collision operator. A modern derivation of such representation is given in [8]}, [9]}, [10]}, [23]}, which revised the Carleman representation in higher dimension ( as is given byThis formula correct a misprint in [10]}, equation (9) and (10).. )
Starting by setting the variables \(z:=-u^+=\frac{1}{2}\left(|u|\sigma +u\right)\)  and \(x:=u^- = \frac{1}{2}\left(|u|\sigma -u\right)\) . This choices makes
to obtain \(-z-u= u^+-\frac{1}{2} u = u^- \) , which implies
\(|u|=|z+x|\) .
In addition, since \(|z|^2= \frac{1}{2}|u|^2(1+ \hat{u}\cdot \sigma )\) , then \(\cos \theta ={\hat{u}\cdot \hat{u^{\prime }}}= 1- 2\tfrac{|z|^2}{|u|^2}= 1- 2\tfrac{|z|^2}{|z+x|^2}, \)  and the \(d\) -dimensional spherical integration element \(d\sigma =|u|^{2-d}d(|u| \sigma +u) = |z+x|^{2-d}dx\) ,
yields the transition probability rate element in the collisional integral is written by
\(B(u,\hat{u}\cdot \sigma ) \frac{1}{|u|^{d-2}} d\sigma = B\left( -(z+x), 1-2\frac{|z|^2}{|z+x|^2} \right) \frac{1}{ |z+x|^{d-2}}dx.\) 
Thus, letting \(g\)  and \(h\)  be a pair suitable functions for which the following integral is well defined, evaluated \(g(v^{\prime }_*)=g(v-u^+)=g(v-x)\)  and \(h(v^{\prime })=h(v+u^+)=h(v+z)\) , the Carleman's representation takes the form,
\(\begin{split}Q^+(g,h) (v) =\ {2^{d-1}}\!\int _{x\in {\mathbb {R}}^d}\frac{g(v+z)}{|x|}\int _{z\cdot x=0} h(v+x)\,\!|z+x|^{2-d}\,B\Big ( -(z+x), 1-2\tfrac{|z|^2}{|z+x|^2} \Big )\,d\pi _z\, dx,\end{split}\) 
where the integral element \(d\pi _z \)  is the Lebesgue measure on the hyperplane \(\lbrace z : \hat{x}\cdot z = 0\rbrace .\)
Using the change of variables \(x\rightarrow x-v\)  yields
\(\begin{split}&Q^{+}(g,h)(v)=2^{d-1}\int _{\mathbb {R}^{d}}\frac{g(x)}{\left|v-x\right|}\int _{(v-x)\cdot z=0}\frac{\tau _{-x}h(z+(v-x))}{\left|z+(v-x)\right|^{d-2}}\,\tilde{a}\left(v,z,x\right) d\pi _z\,dx\\[2pt]&\text{where} \qquad \tilde{a}\left(v,z,x\right) := B\left(|z+(v-x)|,1-2\tfrac{|z|^2}{|z+(v-x)|^2}\right).\end{split}\) 
As noticed in [8]}, applying this representation to the linear form \(Q^{+}(\delta _0,h)\)  one obtains the \(d\) -dimensional weighted Radon transform
\(Q^{+}(\delta _0,h)(v)=\frac{2^{d-1}}{|v|}\int _{v\cdot z=0}\frac{h(z+v)}{|z+v|^{d-2}}\;B\left(|z+v|,1-\frac{2|z|^{2}}{|z+v|^{2}}\right)d\pi _z.\) 
Combining (REF ) and (REF ) we obtain another representation for the collision operator, the double mixing convolution
\(Q^{+}(g,h)(v)=\int _{\mathbb {R}^d}g(x)\,\tau _{x}Q^{+}(\delta _0,\tau _{-x}h)(v)\,dx.\) 
Using this representation, the \(L^2\) -norm of \(Q^{+}\)  can be estimated in terms of the \(L^{2}\) -norm of \(Q^{+}(\delta _0,\cdot )\)  after invoking Minkowski's integral inequality
\(\begin{split}\big \Vert Q^{+}(g,h)\big \Vert _{L^{2}({\mathbb {R}}^{d})}&=\left(\int _{\mathbb {R}^d}\left(\int _{\mathbb {R}^d}g(x)\tau _{x}Q^{+}(\delta _0,\tau _{-x}h)(v)dx\right)^2dv\right)^{1/2}\\&\le \int _{\mathbb {R}^d}\left(\int _{\mathbb {R}^d}\left(\tau _{x}Q^{+}(\delta _0,\tau _{-x}h)(v)\right)^2dv\right)^{1/2}g(x)dx\,.\end{split}\) 

Just as a minor remark, if we define the operator \(\mathcal {P}_o(\cdot ) = \mathcal {P}(\cdot ,1)\) , then \(Q^{+}(\delta _0,\cdot )=\mathcal {P}^{\ast }_o(\cdot )\)  the adjoint of \(\mathcal {P}_o\) .
The following proposition is a slight improvement (in terms of moments) of its analog in [10]}. It shows that adding conditions on \(b\) , such as boundedness, one can improve the Young's inequality of Theorem REF . Other example of this fact can be found in [5]}, [28]}, [29]} that deal with Sobolev regularisation.
Proposition 30 (Gain of integrability for angular transition \(b\in L^\infty (\mathbb {S}^{d-1}) \)  ) 
Let \(h \in L^{1}({\mathbb {R}}^{d}) \cap L^{\frac{2d}{2d-1}}({\mathbb {R}}^{d})\) , with \(d\ge 3\) . Assume cutoff hard sphere potential \(\Phi (u)=|u|\textbf {1}_{|u|\ge \varepsilon }\) , with \(\varepsilon \in (0,1]\) , and with scattering kernel \(b\in L^\infty (\mathbb {S}^{d-1}) \) . Then,
\(\big \Vert Q^{+}_{\Phi }(\delta _0,h)(v)\big \Vert _{L^{2}({\mathbb {R}}^{d})}\le \frac{C_{d}}{\varepsilon ^{\frac{d-3}{2}}}\Vert b\Vert _{\infty }\Vert h\Vert _{L^{\frac{2d}{2d-1}}({\mathbb {R}}^{d})}\,,\) 
for some constant \(C_{d}\)  depending only on the dimension.

Using Radon's representation (REF ) it follows that (for the current potential \(B(u,s) = |u|\textbf {1}_{|u|\ge \varepsilon }\,b(s)\) )
\(Q^{+}_{1}(\delta _0,h)(v)&=\frac{2^{d-1}}{|v|}\int _{z\cdot v=0}\frac{h(z+v)}{|z+v|^{d-2}}\;B\left(|z+v|,1-\frac{2|z|^{2}}{|z+v|^{2}}\right)d\pi _z\\&=\frac{2^{d-1}}{|v|}\int _{z\cdot v=0}\frac{h(z+v)\textbf {1}_{|z+v|\ge \varepsilon }}{|z+v|^{d-3}}\;b\left(1-\frac{2|z|^{2}}{|z+v|^{2}}\right)d\pi _z\\&\le 2^{d-1}\left\Vert b\right\Vert _{\infty }\frac{1}{|v|}\int _{z\cdot v=0}\tilde{h}(z+v)d\pi _z\,,\) 

where \(d\pi _z\)  is the \(\mathbb {R}^{d-1}\)  Lebesgue's measure and \(0\le \tilde{h}(\cdot ) = h(\cdot )\textbf {1}_{|\cdot |\ge \varepsilon }/|\cdot |^{d-3}\) . Using polar coordinates \(v=r\sigma \) , it is possible to estimate its \(L^{2}\) -norm in the following way
\(&(2^{d-1}\left\Vert  b \right\Vert _{\infty })^{-2}\int _{\mathbb {R}^{d}}\big | Q^{+}(\delta _0,h)(v) \big |^{2}dv\\&\hspace{-14.22636pt}\le \int _{ \mathbb {S}^{d-1}} \int _{\mathbb {R}}\int _{z_1\cdot \sigma =0}\int _{z_2\cdot \sigma =0}\tilde{h}(z_1+r\sigma )\tilde{h}(z_2+r\sigma )d\pi _{z_1}d\pi _{z_2}r^{d-3}drd\sigma =:I.\) 

Perform the change of variables, for fixed \(\sigma \) , \(x:=z_1+r\sigma \) . Note that \(r=|x\cdot \sigma |\) , therefore,
\(I=\int _{\mathbb {S}^{d-1}}\int _{\mathbb {R}^{d}}&\int _{z_2\cdot \sigma =0}\tilde{h}(x) \tilde{h}(z_2+(x\cdot \sigma )\sigma )d\pi _{z_2}|x\cdot \sigma |^{d-3}dxd\sigma \\&\le \int _{\mathbb {S}^{d-1}}\int _{\mathbb {R}^{d}}\int _{z_2\cdot \sigma =0}h(x) \tilde{h}(z_2+(x\cdot \sigma )\sigma )d\pi _{z_2}|\hat{x}\cdot \sigma |^{d-3}\textbf {1}_{|x|\ge \varepsilon }dxd\sigma \,.\) 

Writing
\(z_2+(x\cdot \sigma )\sigma =x+(z_2+(x\cdot \sigma )\sigma -x):=x+z_3,\) 
it easy to check that \(z_3\in \lbrace z:z\cdot \sigma =0\rbrace \) . Hence,
\(I\le \int _{\mathbb {S}^{d-1}}\int _{\mathbb {R}^{d}}\int _{z_3\cdot \sigma =0}h(x) \ \tilde{h}(x+z_3)d\pi _{z_3}|\hat{x}\cdot \sigma |^{d-3}\textbf {1}_{|x|\ge \varepsilon } dx d\sigma .\) 
Next, invoking the identity
\(\int _{{\mathbb {R}}^d}\delta _0(z\cdot y)\,\varphi (z)\,dz = |y|^{-1} \int _{z\cdot y=0} \varphi (z)\,d\pi _z \quad \iff \quad \int _{{\mathbb {R}}^d}\delta _0(z\cdot \hat{y})\,\varphi (z)\,dz = \int _{z\cdot \hat{y}=0} \varphi (z)\,d\pi _z\) 
valid for any smooth \(\varphi \)  and \(d\pi _z \)  is the Lebesgue measure on the hyperplane \(\lbrace z : \hat{y}\cdot z = 0\rbrace \) , transforms the integration in the hyperplane
\(\lbrace z_3\cdot \sigma =0\rbrace \)  into an integration in \({\mathbb {R}}^d\) , to get
\(\int _{z_3\cdot \sigma =0}\tilde{h}(x+z_3)d\pi _{z_3}&=\int _{\mathbb {R}^{d}}\delta (z\cdot \sigma )\tilde{h}(x+z)dz\\&=\int _{\mathbb {R}^{d}}\delta (\hat{z}\cdot \sigma )\frac{\tilde{h}(x+z)}{|z|}dz.\) 
Hence,
\(I\le \int _{\mathbb {R}^d}\int _{\mathbb {R}^d}\frac{ h(x) \tilde{h}(x+z)}{|z|}\bigg (\int _{\mathbb {S}^{d-1}}\delta (\hat{z}\cdot \sigma )|\hat{x}\cdot \sigma |^{d-3}d\sigma \bigg )\textbf {1}_{|x|\ge \varepsilon } dx dz\,.\) 
Observing that using polar coordinates with zenith \(\hat{z}\)  it holds that
\(\int _{\mathbb {S}^{d-1}}\delta (\hat{z}\cdot \sigma )(\hat{x}\cdot \sigma )^{d-3}d\sigma \le \int _{\mathbb {S}^{d-1}}\delta (\hat{z}\cdot \sigma )d\sigma =|\mathbb {S}^{d-2}|\,.\) 
Moreover, in the region \(\lbrace |x|\ge \varepsilon \rbrace \)  it follows that \(\tilde{h}\le \varepsilon ^{-(d-3)}h\) . Thus, one concludes that
\(I\le \varepsilon ^{-(d-3)}|\mathbb {S}^{d-2}|\int _{\mathbb {R}^d}\int _{\mathbb {R}^d}\frac{ h(x) h(z)}{|z-x|}dzdx\,.\) 
Invoking Hardy-Littlewood-Sobolev (HLS) inequality, we have for any \(r\in (1,\infty )\) ,
\(I&\le \varepsilon ^{-(d-3)}|\mathbb {S}^{d-2}|\,\Vert h\Vert _{L^{r^{\prime }}({\mathbb {R}}^{d})}\Vert h\ast |x|^{-1}\Vert _{L^{r}({\mathbb {R}}^{d})} \le \varepsilon ^{-(d-3)}C_{d,r,p}\,\Vert h\Vert _{L^{r^{\prime }}({\mathbb {R}}^{d})}\Vert h\Vert _{L^{p}({\mathbb {R}}^{d})},\) 

where \(1/p+1/d=1+1/r\)  and \(C_{d,r,p}\)  is proportional to the HLS constant. Taking the optimal choice \(p=r^{\prime }\) , that is \(p=\frac{2d}{2d-1}\) , yields \(I\le \frac{C_{d}}{\varepsilon ^{d-3}}\Vert h\Vert ^{2}_{L^{\frac{2d}{2d-1}}({\mathbb {R}}^{d})}\) , which is the desired conclusion.
In this way, estimate (REF ) and Proposition REF  readily imply that for \(b\in L^\infty (S^{d-1}) \) ,
\(\big \Vert  Q^{+}_{\Phi }(g,h) \big \Vert _{ L^{2}({\mathbb {R}}^{d}) }\le \tfrac{ C_{d} }{ \varepsilon ^{\frac{d-3}{2}} }\Vert b\Vert _{\infty } \Vert g\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert h\Vert _{L^{\frac{2d}{2d-1}}({\mathbb {R}}^{d})}\,\qquad \Phi (\cdot ) = |\cdot |\textbf {1}_{|\cdot |\ge \varepsilon }\,,\;\; \varepsilon \in (0,1]\,.\) 
Estimate (REF ) is understood as a gain of integrability for the collision operator since the \(L^{2}\) -norm of \(Q^{+}(g,\cdot )\)  is estimated by the lower \(L^{\frac{2d}{2d-1}}\) -norm when considering a fixed integrable first entry \(g\) . One can use Riesz–Thorin interpolation theorem to extend this estimate to the \(L^{p}\) -spaces when combined with the convolution estimates of Theorem REF , see for instance [7]}, [10]}.
Theorem 31 (Gain of integrability) 
Take functions \(g, f\in (L^{1}\cap L^{p})({\mathbb {R}}^{d})\)  with \(p\in (1,\infty )\) . Let \(\tilde{\Phi }(u)=\textbf {1}_{\lbrace |u|\ge \varepsilon \rbrace }\) , with \(\varepsilon \in (0,1]\) , and scattering kernel \(b\in L^\infty (\mathbb {S}^{d-1})\) . Write \(b=b^+ + b^-\)  where
\(b^+(s) = b(s)\text{1}_{\lbrace  s\ge 0 \rbrace  }\qquad \text{and} \qquad b^{-}(s)=b(s)\text{1}_{ \lbrace  s<0 \rbrace }\,,\) 
are the forward and backward scattering components in \(\mathbb {S}^{d-1}_{\pm }\) . Then,
\(\big \Vert Q^{+}_{\tilde{\Phi },b^+}(g,h)(v)\big \Vert _{L^{r}({\mathbb {R}}^{d})}&\le \frac{C}{\varepsilon ^{\theta \frac{d-1}{2}}}\Vert b^+\Vert _{\infty }\Vert g\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert h\Vert _{L^{p}({\mathbb {R}}^{d})}\,,\\\big \Vert Q^{+}_{\tilde{\Phi },b^-}(g,h)(v)\big \Vert _{L^{r}({\mathbb {R}}^{d})}&\le \frac{C}{\varepsilon ^{\theta \frac{d-1}{2}}}\Vert b^-\Vert _{\infty }\Vert h\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert g\Vert _{L^{p}({\mathbb {R}}^{d})}\,,\qquad \theta =\tfrac{2}{\max \lbrace r,r^{\prime }\rbrace }\,,\) 

for some constant \(C\)  depending only on the dimension \(d\ge 3\) . The integral exponents relate as
\(r>p=\left\lbrace \begin{array}{ccl}\frac{r}{r(1-1/d)+1/d} & \text{if} & r \in (1,2] ,\\[0.2cm]\frac{r}{2-1/d} & \text{if} & r \in [2,\infty ) \,.\end{array}\right.\) 
As a consequence,
\(\begin{split}\big \Vert Q^{+}_{\tilde{\Phi },b}(g,h)(v)\big \Vert _{L^{r}({\mathbb {R}}^{d})} &\le \frac{C\,\Vert b\Vert _{\infty }}{\varepsilon ^{\theta \frac{d-1}{2}}}\Big (\Vert g\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert h\Vert _{L^{p}({\mathbb {R}}^{d})} + \Vert h\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert g\Vert _{L^{p}({\mathbb {R}}^{d})}\Big )\,.\end{split}\) 

When \(r\in (1,2]\)  one uses the end point (Theorem REF  for Maxwell interactions with \((p,q,r)=(1,1,1)\) )
\(\big \Vert  Q^{+}_{\tilde{\Phi },b}(g,h) \big \Vert _{ L^{1}({\mathbb {R}}^{d}) } \le \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert g\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert h\Vert _{L^{1}({\mathbb {R}}^{d})}\,,\qquad \tilde{\Phi }= \textbf {1}_{|\cdot |\ge \varepsilon }\,,\;\; \varepsilon \in (0,1]\,.\) 
Interpolating with estimate (REF ), for any \(r\in (1,2]\) , one deduces that (note that \(\tilde{\Phi }\le \varepsilon ^{-1}\Phi \) )
\(\big \Vert  Q^{+}_{\tilde{\Phi },b}(g,h) \big \Vert _{ L^{r}({\mathbb {R}}^{d}) } \le \tfrac{ C }{ \varepsilon ^{\theta \frac{d-1}{2}} }\Vert b\Vert ^{\theta }_{L^{\infty }(\mathbb {S}^{d-1})}\Vert b\Vert ^{1-\theta }_{L^{1}(\mathbb {S}^{d-1})}\Vert g\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert h\Vert _{L^{p}({\mathbb {R}}^{d})} \,,\quad p=\frac{r}{r(1-1/d)+1/d}<r\,.\) 
For the case \(r\in [2,\infty )\)  we use the end point (Theorem REF  for Maxwell interactions with \((p,q,r)=(\infty ,1,\infty )\) )
\(\big \Vert  Q^{+}_{\tilde{\Phi },b^+}(g,h) \big \Vert _{ L^{\infty }({\mathbb {R}}^{d}) } \le C\Vert b^+\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert g\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert h\Vert _{L^{\infty }({\mathbb {R}}^{d})}\,.\) 
Note that \(b^{+}\)  is necessary to keep the constant finite. Interpolation with (REF ) proves that
\(\big \Vert  Q^{+}_{\tilde{\Phi },b^+}(g,h) \big \Vert _{ L^{r}({\mathbb {R}}^{d}) } \le \tfrac{ C }{ \varepsilon ^{\theta \frac{d-1}{2}} }\Vert b^+\Vert ^{\theta }_{L^{\infty }(\mathbb {S}^{d-1})}\Vert b^+\Vert ^{1-\theta }_{L^{1}(\mathbb {S}^{d-1})}\Vert g\Vert _{ L^{1}({\mathbb {R}}^{d}) }\Vert h\Vert _{L^{p}({\mathbb {R}}^{d})} \,,\quad p=\frac{r}{2-1/d}<r\,.\) 
Here the interpolation parameter \(\theta \in (0,1]\)  is given by
\(\theta =\left\lbrace \begin{array}{ccl}\frac{2}{r^{\prime }} & \text{if} & r \in (1,2] ,\\[0.2cm]\frac{2}{r} & \text{if} & r \in [2,\infty ) \,.\end{array}\right.\) 
This proves inequality (REF ) since \(\Vert b^+\Vert _{L^{1}(\mathbb {S}^{d-1})}\le |\mathbb {S}^{d-1}|\,\Vert b^+\Vert _{L^{\infty }(\mathbb {S}^{d-1})}\) . As for inequality () one performs the change \(\sigma \rightarrow -\sigma \)  to notice that
\(Q^{+}_{\tilde{\Phi },b^-}(g,h)(v) = Q^{+}_{\tilde{\Phi },\mathcal {R}b^-}(h,g)(v)\) 
where \((\mathcal {R}b^-)(s)=b^-(-s)\)  is supported in \(\mathbb {S}^{d-1}_+ \) . Then one can apply inequality (REF ) and use that \(\Vert \mathcal {R}b^-\Vert _\infty =\Vert b^{-}\Vert _\infty \) . For the latter inequality (REF ) simply note that
\(\big \Vert Q^{+}_{\tilde{\Phi },b}(g,h)(v)\big \Vert _{L^{r}({\mathbb {R}}^{d})} &= \big \Vert Q^{+}_{\tilde{\Phi },b^+}(g,h)(v)+Q^{+}_{\tilde{\Phi },b^-}(g,h)(v)\big \Vert _{L^{r}({\mathbb {R}}^{d})}\\&\le \big \Vert Q^{+}_{\tilde{\Phi },b^+}(g,h)(v)\big \Vert _{L^{r}({\mathbb {R}}^{d})}+\big \Vert Q^{+}_{\tilde{\Phi },b^-}(g,h)(v)\big \Vert _{L^{r}({\mathbb {R}}^{d})}\,,\) 

and apply inequalities (REF ) and () accordingly.

An energy method: \(L^r\) -propagation theory
Propagation of the \(L^{r}\) -norms for the Boltzmann equation has been of great interest in the community for quite some time, see for instance [4]}, [6]}, [7]}. In [7]} a complete study was presented addressing not only the \(L^r\) -propagation of solutions to the space homogeneous Boltzmann equation, for \(r\in (1,\infty )\) , but also the propagation of Sobolev norms and convergence toward equilibrium.
In this section we use the work done in the previous section, based on [9]} and [10]}, to address the \(L^{r}\) -propagation theory filling along the way some of the issues that remained open in [7]}. We free the scattering kernel \(b\)  of any higher integrability assumption, the only requirement is \(b\in L^{1}(\mathbb {S}^{d-1})\) , and deal with the case \(r=\infty \)  as well. In addition, we present a quite natural argument to find a lower bound for the negative collision operator to clearly address this issue with a minimal requirement on solutions. For a brief comparison between cutoff and non-cutoff \(L^r\) -propagation theories we refer the reader to [39]}.
Throughout the section we use the \(L^{p}_{\ell }\)  spaces which, we recall, are defined as:
\(L^{p}_{\ell }({\mathbb {R}}^d)\ = \ \Big \lbrace  \ g \; \text{measurable} \ : \ \Vert g\Vert _{L^p_\ell } \ = \Vert  \langle \cdot \rangle ^{\ell } g \Vert _{L^{p}} \ < \ \infty \Big \rbrace \,, \qquad p\ge 1,\quad \ell \in \mathbb {R}\,.\) 
Estimating the Dirichlet form of the collision operator for hard potentials 
This subsection focuses on estimates of following from Theorem REF  and Theorem REF  in order to find a suitable estimate for the collision Dirichlet form
\(\int _{\mathbb {R}^d} Q^+(f,f)(v)\,f(v)^{r-1} \langle v \rangle ^{r\,\ell } dv\,,\qquad \ell \ge 0,\quad r\in (1,\infty )\,,\) 
with a transition probability \(B(|u|, \hat{u} \cdot \sigma )= \Phi (|u|) b(\hat{u} \cdot \sigma )\)  with the potential function \(\Phi \)  satisfying conditions from (REF ) and \(b\in L^{1}(\mathbb {S}^{d-1})\) . Such estimates are central when implementing an energy method for the Boltzmann equation. As stated earlier, these ideas started in [6]}, [4]} and were successfully concluded in [7]}. They are inspired in older ideas used for the analysis of the linearized Boltzmann equation where the gain operator is broken down in a big compact (smoothing) part that satisfies a gain of integrability property and a small rough part. Then, one uses the loss part of the collision operator to control such small rough part. In the following argument, we use Theorem REF  which reduces considerably the technicalities of the original arguments given in [7]}.
The main technicality of this new argument is to handle the general case \(b\in L^{1}(\mathbb {S}^{d-1})\) . To this end, one has to make a proper decomposition of the integration domains by writting the collisional kernel as the sum of nonnegative functions as follows
\(|\cdot |^{\gamma } = \Phi (\cdot ) = \Phi _{l}(\cdot ) + \Phi _{s}(\cdot )\,,\qquad b(\cdot ) = b^{\infty }(\cdot ) + b^{1}(\cdot ) \in L^{1}(\mathbb {S}^{d-1})\,.\) 
That is, the component \(b^{\infty }(\mathbb {S}^{d-1}) \in L^{\infty }\)  is a good approximation for \(b\in L^{1}(\mathbb {S}^{d-1})\)  and \(\Phi _{l}(u)\)  relate to large relative velocities of the potential by choosing sufficiently large positive parameters \(R\)  and \(\tilde{a}\)  to make
\(\Phi _{l}(u)=\Phi (u)\,\text{1}_{\lbrace |u| \ge 1/R \rbrace }\qquad \text{and} \qquad \Vert  b^{1} \Vert _{L^{1}(\mathbb {S}^{d-1})} \le \tilde{a}^{-1}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\, ,\) 

in order to estimate
the gain part of collision operator with a collision cross section \(|u|^{\gamma } \, b(\hat{u}\cdot \sigma )\)  as a sum of terms
\(Q^{+}_{\Phi ,b}(f,f) = Q^{+}_{\Phi _{l},b^{\infty }}(f,f) + Q^{+}_{\Phi ,b^{1}}(f,f)+Q^{+}_{\Phi _{s},b^{\infty }}(f,f)\,.\) 
Before estimating each term in the right of (REF ), recall center of mass and relative velocities coordinates introduced in (REF ) to be used in the form
\(u^{+}=\frac{u+|u|\sigma }{2}=v-v^{\prime }_*\qquad \text{and}\qquad u^{-}=\frac{u-|u|\sigma }{2}=v^{\prime }-v\,,\) 
as defined in (REF ). Then the following inequalities hold
\(&{\bf i)} \ \text{If}\ \ \widehat{u}\cdot \sigma \ge 0, \ \text{then} \ |u| = |u^{+}|\sqrt{\tfrac{2}{1+\widehat{u}\cdot \sigma }} =|v-v^{\prime }_*| \sqrt{\tfrac{2}{1+\widehat{u}\cdot \sigma }} \le \langle v^{\prime }_{*} \rangle \langle v \rangle \sqrt{\tfrac{2}{1+\widehat{u}\cdot \sigma }} \le \sqrt{2}\,\langle v^{\prime }_{*} \rangle \, \langle v \rangle \,,\nonumber \\[3pt]&\text{and}\ \\[2pt]&{\bf ii)}\ \text{If}\ \ \widehat{u}\cdot \sigma \le 0, \ \text{then} \ |u| = |u^{-}|\sqrt{\tfrac{2}{1-\widehat{u}\cdot \sigma }} =|v-v^{\prime }| \sqrt{\tfrac{2}{1-\widehat{u}\cdot \sigma }} \le \langle v^{\prime } \rangle \langle v \rangle \sqrt{\tfrac{2}{1-\widehat{u}\cdot \sigma }} \le \sqrt{2}\,\langle v^{\prime } \rangle \, \langle v \rangle \,,\nonumber \ \) 

Thus, from (REF ) it follows for any \(r\in (1,\infty )\)  and \(\frac{1}{r}+\frac{1}{r^{\prime }}=1\)
\(\begin{split}|u|^{\gamma } &= |u^{\prime }|^{\frac{\gamma }{r}}|u|^{\frac{\gamma }{r^{\prime }}} \le 2^{\frac{\gamma }{2r^{\prime }}}\,\langle v^{\prime }_{*} \rangle ^{\gamma }\langle v^{\prime } \rangle ^{\frac{\gamma }{r}}\, \langle v \rangle ^{\frac{\gamma }{r^{\prime }}}\quad \text{ for the case {\bf i)}}\,,\\|u|^{\gamma } &= |u^{\prime }|^{\frac{\gamma }{r}}|u|^{\frac{\gamma }{r^{\prime }}} \le 2^{\frac{\gamma }{2r^{\prime }}}\,\langle v^{\prime } \rangle ^{\gamma }\langle v^{\prime }_* \rangle ^{\frac{\gamma }{r}}\, \langle v \rangle ^{\frac{\gamma }{r^{\prime }}}\quad \text{ for the case {\bf ii)}}\,.\end{split}\) 

Further, write \(b^\infty = b^\infty _{+} + b^\infty _{-}\)  the forward and backward scattering components supported in \(\mathbb {S}^{d-1}_{\pm }\)  respectively, then from (REF ) it holds that
\(Q^{+}_{\Phi _{l},b^{\infty }_+}(f,f)(v)\le 2^{\frac{\gamma }{2r^{\prime }}}\,\langle v \rangle ^{\frac{\gamma }{r^{\prime }}}\,Q^{+}_{\tilde{\Phi },b^{\infty }_+}(\langle \cdot \rangle ^{\gamma }\,f,\langle \cdot \rangle ^{\frac{\gamma }{r}}\,f)(v) \nonumber \\Q^{+}_{\Phi _{l},b^{\infty }_-}(f,f)(v)\le 2^{\frac{\gamma }{2r^{\prime }}}\,\langle v \rangle ^{\frac{\gamma }{r^{\prime }}}\, Q^{+}_{\tilde{\Phi },b^{\infty }_-}(\langle \cdot \rangle ^{\frac{\gamma }{r}}\,f,\langle \cdot \rangle ^{\gamma }\,f)(v)\) 

Thus, the corresponding Dirichlet form for the first term of (REF ) is estimated as follows: using Theorem REF  inequality (REF ) and standard Lebesgue's interpolation one has
\(\begin{aligned}\int _{\mathbb {R}^{d}}Q^{+}_{\Phi _{l},b^{\infty }_+}(f,f)(v)&\,f(v)^{r-1} dv\le 2^{\frac{\gamma }{2r^{\prime }}}\Vert Q^{+}_{\tilde{\Phi },b^{\infty }_+}\big ( \langle \cdot \rangle ^{\gamma }\,f,\langle \cdot \rangle ^{\frac{\gamma }{r} }f \big )\Vert _{L^{r}(\mathbb {R}^{d})}\Vert  f \Vert ^{r-1}_{L^{r}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\\&\le C_{R,b^{\infty }_+}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert _{L^{p}_{\frac{\gamma }{r}}(\mathbb {R}^{d})}\Vert  f \Vert ^{r-1}_{L^{r}_{{\frac{\gamma }{r} }}(\mathbb {R}^{d})}\\&\le C_{R,b^{\infty }_+}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert ^{1-\theta _r}_{L^{1}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\,,\qquad \theta _r\in (0,1)\,.\end{aligned}\) 
Similarly, Theorem REF  inequality () renders
\(\begin{aligned}\int _{\mathbb {R}^{d}}Q^{+}_{\Phi _{l},b^{\infty }_-}(f,f)(v)&\,f(v)^{r-1} dv\le 2^{\frac{\gamma }{2r^{\prime }}}\Vert Q^{+}_{\tilde{\Phi },b^{\infty }_-}\big ( \langle \cdot \rangle ^{\frac{\gamma }{r}}\,f,\langle \cdot \rangle ^{\gamma }f \big )\Vert _{L^{r}(\mathbb {R}^{d})}\Vert  f \Vert ^{r-1}_{L^{r}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\\&\le C_{R,b^{\infty }_-}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert _{L^{p}_{\frac{\gamma }{r}}(\mathbb {R}^{d})}\Vert  f \Vert ^{r-1}_{L^{r}_{{\frac{\gamma }{r} }}(\mathbb {R}^{d})}\\&\le C_{R,b^{\infty }_-}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert ^{1-\theta _r}_{L^{1}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\,,\qquad \theta _r\in (0,1)\,.\end{aligned}\) 
Here the constants \(C_{R,b^{\infty }_\pm } = C\, R^{ \frac{d-1}{\max \lbrace r,r^{\prime }\rbrace } }\;\Vert  b^{\infty }_\pm \Vert _{ L^{\infty } }\)  blows up as \(R\)  and \(\tilde{a}\)  increase. Writing
\(Q^{+}_{\Phi _{l},b^{\infty }}(f,f)(v) = Q^{+}_{\Phi _{l},b^{\infty }_+}(f,f)(v) + Q^{+}_{\Phi _{l},b^{\infty }_-}(f,f)(v)\,,\) 
one uses estimates (REF ) and (REF ) to conclude that
\(\int _{\mathbb {R}^{d}}Q^{+}_{\Phi _{l},b^{\infty }}(f,f)(v)\,f(v)^{r-1} dv \le C_{R,b^{\infty }}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert ^{1-\theta _r}_{L^{1}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\,,\qquad \theta _r\in (0,1)\,.\) 
The second term in (REF ) is estimated in a similar manner using Theorem REF  instead of Theorem REF . Writing, as before, \(b^1 = b^1_+ + b^1_{-}\)  the forward and backward scattering components supported in \(\mathbb {S}^{d-1}_\pm \)  respectively, it follows that
\(\begin{aligned}\int _{\mathbb {R}^{d}}Q^{+}_{\Phi ,b^{1}_+}(f,f)(v)f(v)^{r-1}dv &\le 2^{\frac{\gamma }{2r^{\prime }}}\Vert  Q^{+}_{b^{1}_+}\big (\langle \cdot \rangle ^{\gamma } f,\langle \cdot \rangle ^{\frac{\gamma }{r}}f\big )\Vert _{L^{r}(\mathbb {R}^{d})}\Vert  f \Vert ^{r-1}_{L^{r}_{\frac{\gamma }{r}}(\mathbb {R}^{d})} \\&\le C\Vert b^{1}_+\Vert _{L^{1}(\mathbb {S}^{d-1})} \,\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}}(\mathbb {R}^{d})}\\&\le C\,\tilde{a}^{-1}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}}(\mathbb {R}^{d})}\,.\end{aligned}\) 
Similarly,
\(\begin{aligned}\int _{\mathbb {R}^{d}}Q^{+}_{\Phi ,b^{1}_-}(f,f)(v)f(v)^{r-1}dv &\le 2^{\frac{\gamma }{2r^{\prime }}}\Vert  Q^{+}_{b^{1}_-}\big (\langle \cdot \rangle ^{\frac{\gamma }{r}} f,\langle \cdot \rangle ^{\gamma }f\big )\Vert _{L^{r}(\mathbb {R}^{d})}\Vert  f \Vert ^{r-1}_{L^{r}_{\frac{\gamma }{r}}(\mathbb {R}^{d})} \\&\le C\Vert b^{1}_-\Vert _{L^{1}(\mathbb {S}^{d-1})} \,\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}}(\mathbb {R}^{d})}\\&\le C\,\tilde{a}^{-1}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}}(\mathbb {R}^{d})}\,.\end{aligned}\) 
Consequently, from inequalities (REF ) and (REF ) and since
\(Q^{+}_{\Phi ,b^{1}}(f,f)(v)=Q^{+}_{\Phi ,b^{1}_+}(f,f)(v)+Q^{+}_{\Phi ,b^{1}_-}(f,f)(v)\,,\) 
it holds that
\(\int _{\mathbb {R}^{d}}Q^{+}_{\Phi ,b^{1}}(f,f)(v)f(v)^{r-1}dv \le C\,\tilde{a}^{-1}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}}(\mathbb {R}^{d})}\,.\) 
For the latter term in (REF ) simply note, since \(\Phi _{s}(u)=|u|^{\gamma }\text{1}_{\lbrace  |u|<1/R \rbrace  }\le R^{-\gamma }\) , that
\(Q^{+}_{\Phi _{s},b^{\infty }}(f,f)\le R^{-\gamma }Q^{+}_{b^{\infty }}(f,f)\,.\) 
Therefore, by exploiting the quadratic nature of the estimate, Theorem REF  used in \(b^{\infty }_{\pm }\)  readily gives
\(\int _{\mathbb {R}^{d}}Q^{+}_{\Phi _{s},b^{\infty }}(f,f)(v) f(v)^{r-1}dv \le C\,R^{-\gamma }\,\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\,\Vert f\Vert _{L^{1}(\mathbb {R}^{d})}\Vert f\Vert ^{r}_{L^{r}(\mathbb {R}^{d})}\,.\) 
Gathering estimates (REF ), (REF ), and (REF ), one has
\(\int _{\mathbb {R}^{d}}Q^{+}(f,f)(v) f(v)^{r-1}dv \le C\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Big ( \big ( \tilde{a}^{-1} + R^{-\gamma }\big )&\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}}(\mathbb {R}^{d})} \\& + C_{R,b^{\infty }}\Vert  f \Vert ^{1-\theta _r}_{L^{1}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} }(\mathbb {R}^{d})}\Big )\,.\) 

Adding the weight \(\langle \cdot \rangle ^{r\,\ell }\)  in the Dirichlet form is not a problem after using the inequality
\(\langle v \rangle ^{r\,\ell }=\langle v \rangle ^{\ell }\langle v \rangle ^{(r-1)\,\ell }\le \langle v^{\prime }_{*} \rangle ^{\ell }\langle v^{\prime } \rangle ^{\ell }\langle v \rangle ^{(r-1)\,\ell }\,.\) 
Consequently,
\(\int _{\mathbb {R}^{d}}&Q^{+}(f,f)(v) f(v)^{r-1}\langle v \rangle ^{r\ell }dv \le \int _{\mathbb {R}^{d}}Q^{+}\big (\langle \cdot \rangle ^{\ell }f,\langle \cdot \rangle ^{\ell }f\big )(v) \big ( f(v)\langle v \rangle ^{\ell } \big )^{r-1}dv \\&\le C\Vert  f \Vert _{L^{1}_{\gamma +\ell }(\mathbb {R}^{d})}\Big ( \big ( \tilde{a}^{-1} + R^{-\gamma }\big )\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}+\ell }(\mathbb {R}^{d})} + C_{R,b^{\infty }}\Vert  f \Vert ^{1-\theta _r}_{L^{1}_{\frac{\gamma }{r} +\ell }(\mathbb {R}^{d})}\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} +\ell }(\mathbb {R}^{d})}\Big )\,.\) 

Furthermore, if \(f\)  satisfies the condition of Lemma REF  one has that
\(\int _{\mathbb {R}^{d}}Q^{-}(f,f)(v)\langle v \rangle ^{r\,\ell }f(v)^{r-1}dv &= \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\int _{\mathbb {R}^{d}}\big (f(v)\langle v \rangle ^{k}\big )^{r} \big (f\ast |\cdot |^{\gamma }\big )(v)dv \\&\ge c_{lb}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\int _{{\mathbb {R}}^{d}}\big (f(v)\langle v \rangle ^{\frac{\gamma }{r} + \ell }\big )^{r}dv = c_{lb}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{2} + \ell }({\mathbb {R}}^{d})}\,.\) 

As a consequence, adding these latter two estimates we obtain, for any \(R>0\) , \(\tilde{a}>0\)  and \(r\in (1,\infty )\) , that
\(\begin{split}\int _{\mathbb {R}^{d}} Q(f,f)(v) \langle v \rangle ^{r\,\ell }f(v)^{r-1}dv &\le C\,\Vert  f \Vert _{L^{1}_{\gamma +\ell }(\mathbb {R}^{d})}\Big ( \big ( \tilde{a}^{-1} + R^{-\gamma }\big )\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}+\ell }(\mathbb {R}^{d})} \\&\qquad + C_{R,b^\infty }\Vert  f \Vert ^{1-\theta _r}_{L^{1}_{\frac{\gamma }{r} +\ell }(\mathbb {R}^{d})}\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} +\ell }(\mathbb {R}^{d})}\Big ) - c_{lb}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{2} + \ell }({\mathbb {R}}^{d})}\,.\end{split}\) 

Before entering in the specifics of the statement of the following theorem, based on previous discussion, we mention the constant
\(C_{\ell ,\alpha }:=\sup _{t\ge 0}\Vert  f(t) \Vert _{L^{1}_{\alpha +\ell }(\mathbb {R}^{d})}(t) = \sup _{t\ge 0} m_{\frac{\alpha +\ell }{2}}(t) \le \max \Big \lbrace  m_{\frac{\alpha +\ell }{2}}[f _0],\, \mathbb {E}_{\frac{\alpha +\ell }{2}}\Big \rbrace <\infty \,,\quad \alpha \ge 0\,,\quad \ell \ge 0\,,\) 
with the \((\alpha +\ell )/2\) -moments globally controlled by estimates (REF ) and (REF ) developed in Theorem REF .
Theorem 32 (Hard potentials) 
Let \(b\in L^{1}(\mathbb {S}^{d-1})\) , \(\gamma \in (0,1]\) , \(\ell \ge 0\) , and \(r\in (1,\infty )\) . Assume that \(f\)  satisfies the hypothesis of the Lower Bound Lemma REF  and \(m_{\frac{\gamma +\ell }{2}}[f _0]<\infty \) . Then, there exists a constant \(\bar{C}_\gamma :=\bar{C}_\gamma (\Vert  b\Vert _1 ,C_{\ell ,\gamma })\) , estimated below in (REF ) and (REF ), such that
\(\int _{\mathbb {R}^{d}} Q(f,f)(v) \langle v \rangle ^{r\,\ell }f(v)^{r-1}dv &\le \bar{C}_\gamma - \tfrac{c_{lb}}{2}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{2} + \ell }({\mathbb {R}}^{d})} \,, \qquad r>1,\) 

with \(C_{\ell , \gamma }\)  defined in (REF ) controlling the propagation of moments estimates (REF )and (REF ) developed in Theorem REF  and \(c_{lb}\)  was given in Lemma REF .

Using the hypothesis, estimate (REF ) reduces to
\(\int _{\mathbb {R}^{d}} Q(f,f)(v) \langle v \rangle ^{r\,\ell }f(v)^{r-1}dv &\le C_{\ell ,\gamma }\Big ( \big ( \tilde{a}^{-1} + R^{-\gamma }\big )\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{r}+\ell }(\mathbb {R}^{d})} \\&\qquad + C_{R,b^\infty }\,C^{1-\theta _r}_{\ell ,\gamma }\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} +\ell }(\mathbb {R}^{d})}\Big ) - c_{lb}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{2} + \ell }({\mathbb {R}}^{d})}\\& \le C_{R,b^\infty }\,C^{1-\theta _r}_{\ell ,\gamma }\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} +\ell }(\mathbb {R}^{d})} - \tfrac{3c_{lb}}{4}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{2} + \ell }({\mathbb {R}}^{d})}\,,\) 

where, in the last inequality, the parameters \(R\)  and \(\tilde{a}\)  are chosen sufficiently large so that \(C_{\ell ,\gamma }\big ( \tilde{a}^{-1} + R^{-\gamma }\big )\le \frac{c_{lb}}{4}\) . For instance, we can choose
\(\tilde{a}=\frac{8\,C_{\ell ,\gamma }}{c_{lb}} \,,\quad \text{and then} \quad R=\tilde{a}^{1/\gamma } = \Big (\frac{8\,C_{\ell ,\gamma }}{c_{lb}}\Big )^{1/\gamma }\,.\) 
Furthermore, since \(\theta _{r}\in (0,1)\) , we can use Young's inequality controlling
\(C_{R,b^\infty }\,C^{2-\theta _r}_{\ell ,\gamma }\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} +\ell }}\le \Big (\frac{4}{c_{lb}\Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}}\Big )^{\frac{r+\theta _{r}-1}{1-\theta _{r}}}\big (C_{R,b^\infty }\,C^{2-\theta _r}_{\ell ,\gamma }\big )^{\frac{r}{1-\theta _{r}}} + \tfrac{c_{lb}}{4}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})} \Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{2} + \ell }},\) 
to obtain that
\(C_{R,b^\infty }\,C^{2-\theta _r}_{\ell ,\gamma }\Vert  f \Vert ^{r+\theta _r - 1}_{L^{r}_{\frac{\gamma }{r} +\ell }(\mathbb {R}^{d})} - \tfrac{3c_{lb}}{4}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{2} + \ell }({\mathbb {R}}^{d})} \\ \le \Big (\frac{4}{c_{lb}\Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}}\Big )^{\frac{r+\theta _{r}-1}{1-\theta _{r}}}\big (C_{R,b^\infty }\,C^{2-\theta _r}_{\ell ,\gamma }\big )^{\frac{r}{1-\theta _{r}}} - \tfrac{c_{lb}}{4}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\frac{\gamma }{2} + \ell }({\mathbb {R}}^{d})}\,.\) 
Recalling that \(C_{R,b^\infty } = C\, R^{ \frac{d-1}{\max \lbrace r,r^{\prime }\rbrace } }\;\Vert  b^{\infty } \Vert _{ L^{\infty } }\) , we can rewrite the first term in the right hand side as
\(\bar{C}_{\gamma } = \bar{C}_{\gamma }(b, r , C_{\ell ,\gamma }) := \Vert  b \Vert ^{1-\frac{r}{1-\theta _{r}}}_{L^{1}(\mathbb {S}^{d-1})}\Vert  b^{\infty } \Vert ^{\frac{r}{1-\theta _{r}}}_{ L^{\infty } }\Big (\frac{4}{c_{lb} }\Big )^{ \frac{r+\theta _{r}-1}{1-\theta _{r}}}\Big (C\,R^{ \frac{d-1}{\max \lbrace r,r^{\prime }\rbrace } }\,C^{2-\theta _r}_{\ell ,\gamma }\Big )^{\frac{r}{1-\theta _{r}}}\,,\) 
which proves the result.

The Maxwell molecules case
Note that in order to control \(Q^{+}_{\Phi _{s},b^{\infty }}\) , the small relative velocities, the kinetic potential \(\Phi (u)\)  needs to vanish as \(|u|\rightarrow 0\) . This is not the case for the Maxwell Molecules case. Indeed, Maxwell molecules is a critical case for uniform propagation of \(L^{r}\) -integrability for general initial data in \(L^{r}\)  in the sense that as soon as \(\gamma <0\) , i.e. soft potential case, the uniform propagation of moments is lost for general initial data as discussed in [44]} let alone the \(L^{r}_{\ell }\) -norms For the non-cutoff Boltzmann problem this is not the case and the analysis is quite different.. In order to compensate for this criticality issue, we will use the additional compactness gained through propagation of entropy after noticing that we implicitly have the additional requirement on the initial data
\(\int _{\mathbb {R}^{d}}f_0(v)\ln \big (f_0(v)\big )dv<\infty \,.\) 
In our context \(f_0\in \big (L^{1}_{2}\cap L^{r}\big )(\mathbb {R}^{d})\)  for \(r>1\) . As a consequence, the initial entropy is finite since for any \(1<r<\frac{d}{d-1}\le 2\)
\(\bigg |\int _{\mathbb {R}^{d}} &f_0(v)\ln \big (f_0(v)\big )dv \bigg | \le \int _{\lbrace f_0\le 1\rbrace } \big |f_0(v)\ln \big (f_0(v)\big )\big |dv + \int _{\lbrace f_0\ge 1\rbrace } |f_0(v)\ln \big (f_0(v)\big )| dv\\& \le C_{r}\int _{\lbrace f_0\le 1\rbrace } \big |f_0(v)\big |^{\frac{1}{r}} dv + \int _{\lbrace f_0\ge 1\rbrace } |f_0(v)|^{r} dv\\& \le C_{r,d}\bigg (\int _{\lbrace f_0\le 1\rbrace } f_0(v)\langle v \rangle ^{r} dv\bigg )^{\frac{1}{r}} + \int _{\lbrace f_0\ge 1\rbrace } |f_0(v)|^{r} dv =: C\Big (d,\Vert f_0\Vert _{(L^{1}_{2}\cap L^{r})(\mathbb {R}^{d})}\Big )<+\infty .\) 

In the second estimate we used that \(|x\ln (x)| \le C_{p}\,x^{\frac{1}{r}}\) , for any \(r>1\)  and \(x\in [0,1]\) . Furthermore, it is well known that using energy conservation and entropy propagation it follows for the solution of the homogeneous Boltzmann equation \(f(v,t)\) , see [45]} or more recently [46]}, that
\(\sup _{t\ge 0}\int _{\mathbb {R}^{d}}f(v,t)\big |\ln \big (f(v,t)\big )\big | dv \le C\bigg (\int f_0\ln (f_0)dv,\int f_0| v |^{2}dv\bigg )=:C_{\mathcal {M}}\,.\) 
Theorem 33 (Maxwell Molecules) 
Let \(b\in L^{1}(\mathbb {S}^{d-1})\) . Fix \(\ell \ge 0\)  and assume that
\(\Vert  f \Vert _{L^{1}_{\ell }(\mathbb {R}^{d})} = C_{\ell ,0} < \infty \,, \qquad \int _{\mathbb {R}^{d}}f(v)\big |\ln \big (f(v)\big )\big | dv = C_{\mathcal {M}} < \infty \,.\) 
Then, there exists a constant \(\bar{C}_{0}:=\bar{C}_{0}(\Vert  b\Vert _1,C_{\ell ,0},C_{\mathcal {M}})\) , estimated below in (REF ),(REF ) and (REF ), such that
\(\int _{\mathbb {R}^{d}} Q_0(f,f)(v) \langle v \rangle ^{r\,\ell }f(v)^{r-1}dv &\le \bar{C}_{0} - \tfrac{1}{2}\Vert b\Vert _{L^1(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\ell }({\mathbb {R}}^{d})}\,,\quad \forall \,r\in [1,\infty )\,.\) 

Use the decomposition \(b(s)=b(s)\text{1}_{\lbrace |s|<1-\varepsilon \rbrace } + b(s)\text{1}_{\lbrace |s|\ge 1-\varepsilon \rbrace }=:b_{1}^{\varepsilon }+b_{2}^{\varepsilon }\) . Since \(\Vert  b_{2}^{\varepsilon }\Vert _{L^{1}(\mathbb {S}^{d-1})}\)  decreases monotonically to zero with \(\varepsilon \in (0,1)\) , we can choose \(\varepsilon :=\varepsilon (R)\)  sufficiently small such that
\(\Vert  b_{2}^{\varepsilon }\Vert _{L^{1}(\mathbb {S}^{d-1})}\le R^{-1}\Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}\,,\qquad R>0\,.\) 
Write \(Q^{+}_{0,b}(f,f)= Q^{+}_{0,b^{\varepsilon }_{1}}(f,f) + Q^{+}_{0,b^{\varepsilon }_{2}}(f,f)\) .
On the one hand invoking Theorem REF , used in \(b^{\varepsilon }_{2,\pm }\)  the upper and lower scattering of \(b^{\varepsilon }_{2}\)  exploiting the quadratic nature of the estimate, and the fact that \(\langle v \rangle \le \langle v^{\prime }_* \rangle \langle v^{\prime } \rangle \)  we find that
\(\Vert  Q^{+}_{0,b^{\varepsilon }_{2}}(f,f) \Vert _{L^{r}_{k}(\mathbb {R}^{d})} &\le \Vert  Q^{+}_{0,b^{\varepsilon }_{2}}(\langle \cdot \rangle ^{\ell }f,\langle \cdot \rangle ^{\ell }f) \Vert _{L^{r}(\mathbb {R}^{d})} \le 2^{\frac{d}{2r^{\prime }}+1}\Vert  b^{\varepsilon }_{2} \Vert _{L^{1}(\mathbb {S}^{d-1})} \Vert  f \Vert _{L^{1}_{\ell }(\mathbb {R}^{d})}\Vert  f \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})}\\&\le 2^{\frac{d}{2r^{\prime }}+1}R^{-1}\Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})} \Vert  f \Vert _{L^{1}_{\ell }(\mathbb {R}^{d})}\Vert  f \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})}\,.\) 

On the other hand, the inequality \(\langle v \rangle \le \langle v^{\prime }_* \rangle + \langle v^{\prime } \rangle \)  lead us to
\(Q^{+}_{0,b^{\varepsilon }_{1}}(f,f)\,\langle v \rangle ^{\ell } \le 2^{\ell -1}Q^{+}_{0,b^{\varepsilon }_{1}}(f,\langle \cdot \rangle ^\ell f) + 2^{k-1}Q^{+}_{0,b^{\varepsilon }_{1}}(\langle \cdot \rangle ^\ell f,f)\,.\) 
Let us start with the first term in the right. Note that for \(K>1\)
\(Q^{+}_{0,b^{\varepsilon }_{1}}(f,\langle \cdot \rangle ^{\ell } f) &= Q^{+}_{0,b^{\varepsilon }_{1}}(f\,\text{1}_{\lbrace f\le K\rbrace },\langle \cdot \rangle ^{\ell }f) + Q^{+}_{0,b^{\varepsilon }_{1}}(f\,\text{1}_{\lbrace f > K\rbrace },\langle \cdot \rangle ^{\ell }f)\\&\le Q^{+}_{0,b^{\varepsilon }_{1}}(f\,\text{1}_{\lbrace f\le K\rbrace },\langle \cdot \rangle ^{\ell }f) + \ln (K)^{-1}\,Q^{+}_{0,b^{\varepsilon }_{1}}(f\,\ln [f],\langle \cdot \rangle ^{\ell }f)\,.\) 

Using again Theorem REF , one concludes that
\(\Vert  Q^{+}_{0,b^{\varepsilon }_{1}}(f\,\ln [f],\langle \cdot \rangle ^{\ell }f) \Vert _{L^{r}(\mathbb {R}^{d})} &\le \varepsilon ^{-\frac{d}{2r^{\prime }}}\,2^{\frac{d}{2r^{\prime }}} \Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert f\,\ln [f]\Vert _{L^{1}(\mathbb {R}^{d})}\Vert  f \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})} \,,\\\text{and}\qquad \Vert  Q^{+}_{0,b^{\varepsilon }_{1}}(f\,\text{1}_{\lbrace f\le K\rbrace },\langle \cdot \rangle ^{\ell }f) \Vert _{L^{r}(\mathbb {R}^{d})} &\le \varepsilon ^{-\frac{d}{2r^{\prime }}}\,2^{\frac{d}{2r^{\prime }}} \Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert f\,\text{1}_{\lbrace f\le K\rbrace }\Vert _{L^{\frac{2r}{r+1}}(\mathbb {R}^{d})}\Vert \langle \cdot \rangle ^{\ell } f\Vert _{L^{\frac{2r}{r+1}}(\mathbb {R}^{d})}\\&\le \varepsilon ^{-\frac{d}{2r^{\prime }}}\,K^{\frac{1}{2r^{\prime }}}\,2^{\frac{d}{2r^{\prime }}} \Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert f\Vert ^{\frac{r+1}{2r}}_{L^{1}(\mathbb {R}^{d})}\Vert  f\Vert _{L^{\frac{2r}{r+1}}_{\ell }(\mathbb {R}^{d})}\,.\) 

Use, in the latter inequality, the interpolation
\(\Vert  f \Vert _{L^{\frac{2r}{r+1}}_{\ell }(\mathbb {R}^{d})}\le \Vert  f \Vert ^{\frac{1}{2}}_{L^{1}_{\ell }(\mathbb {R}^{d})}\Vert  f \Vert ^{\frac{1}{2}}_{L^{r}_{\ell }(\mathbb {R}^{d})}\,,\) 
and gather the estimates to conclude that
\(\Vert  Q^{+}_{0,b^{\varepsilon }_{1}}(f,\langle \cdot \rangle ^\ell f) \Vert _{L^{r}(\mathbb {R}^{d})} \le \varepsilon ^{-\frac{d}{2r^{\prime }}}\,2^{\frac{d}{2r^{\prime }}}\,\Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}\Big (\ln (K)^{-1} & \Vert  f \ln [f] \Vert _{L^{1}(\mathbb {R}^{d}) } \Vert  f \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})} \\&+ K^{\frac{1}{2r^{\prime }}}\,\Vert  f \Vert ^{1+\frac{1}{2r}}_{L^{1}_{\ell }(\mathbb {R}^{d}) }\Vert  f \Vert ^{\frac{1}{2}}_{L^{r}_{\ell }(\mathbb {R}^{d})}\Big )\,.\) 

Since \(Q^{+}_{0,b^{\varepsilon }_{1}}(\langle \cdot \rangle ^{\ell }f,f)=Q^{+}_{0,\mathcal {R}b^{\varepsilon }_{1}}(f,\langle \cdot \rangle ^{\ell }f)\)  an identical estimate follows for the second term
\(Q^{+}_{0,b^{\varepsilon }_{1}}(\langle \cdot \rangle ^\ell f, f) \le \varepsilon ^{-\frac{d}{2r^{\prime }}}\,2^{\frac{d}{2r^{\prime }}}\, \Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}\Big (\ln (K)^{-1} \Vert f \ln [f] \Vert _{L^{1}(\mathbb {R}^{d}) } & \Vert  f \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})} \\&+ \,K^{\frac{1}{2r^{\prime }}}\,\Vert  f \Vert ^{1+\frac{1}{2r}}_{L^{1}_{\ell }(\mathbb {R}^{d}) }\Vert  f \Vert ^{\frac{1}{2}}_{L^{r}_{\ell }(\mathbb {R}^{d})}\Big )\,.\) 

Overall, we are led to the estimate
\(\begin{split}\Vert Q^{+}_{0,b}(f,f)\Vert _{ L^{r}_{\ell }(\mathbb {R}^{d}) }&\le 2^{\frac{d}{2r^{\prime }}}\Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}\Big ( \ln (K)^{-1}2^{\ell -1}\varepsilon ^{-\frac{d}{2r^{\prime }}}\Vert f \ln [f] \Vert _{L^{1}(\mathbb {R}^{d}) } \Vert  f \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})} \\&\qquad + 2^{\ell -1}\varepsilon ^{-\frac{d}{2r^{\prime }}}\,K^{\frac{1}{2r^{\prime }}}\,\Vert  f \Vert ^{1+\frac{1}{2r}}_{L^{1}_{\ell }(\mathbb {R}^{d}) }\Vert  f \Vert ^{\frac{1}{2}}_{L^{r}_{\ell }(\mathbb {R}^{d})} + R^{-1}\Vert  f \Vert _{L^{1}_{\ell }(\mathbb {R}^{d}) } \Vert  f \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})}\Big )\\&= 2^{\frac{d}{2r^{\prime }}}\Vert  b \Vert _{L^{1}(\mathbb {S}^{d-1})}\Big (2^{\ell -1}\varepsilon ^{-\frac{d}{2r^{\prime }}}\,K^{\frac{1}{2r^{\prime }}}\,C^{1+\frac{1}{2r}}_{\ell }\Vert  f \Vert ^{\frac{1}{2}}_{L^{r}_{\ell }(\mathbb {R}^{d})} \\&\hspace{71.13188pt} + \big (2^{\ell -1}\varepsilon ^{-\frac{d}{2r^{\prime }}}\,C_{\mathcal {M}}\,\ln (K)^{-1} + C_{\ell ,0}R^{-1}\big )\Vert  f \Vert _{L^{r}_\ell (\mathbb {R}^{d})}\Big )\,.\end{split}\) 

With this estimate at hand, the upper control goes as follows. Using Hölder's inequality one notices that
\(\begin{split}\int _{\mathbb {R}^{d}}Q_0(f,f)(v)\langle v \rangle ^{r\,\ell }&f^{r-1}(v) dv \\&= \int _{\mathbb {R}^{d}}Q^{+}_0(f,f)(v)\langle v \rangle ^{\ell }\big ( \langle \cdot \rangle ^{\ell }f \big )^{r-1}(v) dv - \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\ell }(\mathbb {R}^{d})} \\& \le \Vert Q^{+}_0(f,f) \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})}\Vert  f\Vert ^{r-1}_{L^{r}_{\ell }(\mathbb {R}^{d})} - \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert ^{r}_{L^{r}_{\ell }(\mathbb {R}^{d})}\,.\end{split}\) 

Taking first \(R>0\)  and then \(K>1\)  sufficiently large in (REF ) such that
\(2^{\frac{d}{2r^{\prime }}}\, 2^{\ell -1} \, \varepsilon (R)^{-\frac{d}{2r^{\prime }}} \, C_{\mathcal {M}}\, \ln (K)^{-1} + 2^{\frac{d}{2r^{\prime }}}\, C_{\ell ,0}\, R^{-1}\le \frac{1}{4}\, ,\) 
for example one can take
\(R:= 2^{\frac{d}{2r^{\prime }}}\,8\,C_{\ell ,0}\,,\quad \text{and then}\quad K:=\exp \Big (\frac{2^{\frac{d}{2r^{\prime }}}\,2^{\ell -1}\,8\,C_{\mathcal {M}}}{\varepsilon (R)^{\frac{d}{2r^{\prime }}}}\Big )\,,\) 
and plugging into (REF ) we readily obtain, thanks to Young's inequality, that
\(\int _{\mathbb {R}^{d}}Q(f,f)(v)\langle v \rangle ^{r\,\ell }f^{r-1}(v)dv &\le \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Big (2^{\ell -1}\,2^{\frac{d}{2r^{\prime }}}\,\varepsilon ^{-\frac{d}{2r^{\prime }}}\,K^{\frac{1}{2r^{\prime }}}\,C^{1+\frac{1}{2r}}_{\ell , 0}\Vert f\Vert ^{r-\frac{1}{2}}_{L^{r}(\mathbb {R}^{d})} - \tfrac{3}{4}\Vert  f \Vert ^{r}_{L^{r}_{\ell }(\mathbb {R}^{d})}\Big )\\&\le \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Big (\Big (4^{1-\frac{1}{2r}}\,2^{\ell -1}\,2^{\frac{d}{2r^{\prime }}}\,\varepsilon ^{-\frac{d}{2r^{\prime }}}\,K^{\frac{1}{2r^{\prime }}}\,C_{\ell ,0}^{1+\frac{1}{2r}}\Big )^{2r} - \tfrac{1}{2}\Vert  f \Vert ^{r}_{L^{r}_{\ell }(\mathbb {R}^{d})}\Big )\,.\) 

We can choose
\(\bar{C}_{0}:=\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Big (4^{1-\frac{1}{2r}}\,2^{\ell -1}\,2^{\frac{d}{2r^{\prime }}}\,\varepsilon (R)^{-\frac{d}{2r^{\prime }}}\,K^{\frac{1}{2r^{\prime }}}\,C_{\ell ,0}^{1+\frac{1}{2r}}\Big )^{2r}\) 
which proves the result.

Propagation of \(L^{r}_{\ell }\)  norms.
The conditions of Theorem REF  and Theorem REF  hold for any solution \(f(t)\)  of the Boltzmann equation with finite mass and energy. Additionally, the solutions' entropy is uniformly bounded as explained in (REF ) which is an important fact when addressing the case of Maxwell molecules. Thus, multiplying the Boltzmann equation (REF ) by \(\langle v \rangle ^{r\,\ell }f(v)^{r-1}\) , integrating in velocity, and using Theorem REF  for the case of hard potentials, after bounding \(\Vert  f(t) \Vert _{L^{r}_{ \frac{\gamma }{2}+{\ell }}}\ge \Vert  f(t) \Vert _{L^{r}_{\ell }}\) ; or Theorem REF  for the case of Maxwell molecules, it follows the linear ordinary differential inequality
\(\frac{d X}{dt} \le \bar{C}_{\gamma } - \tfrac{c_{lb}}{2}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\, X\,, \qquad \quad \gamma \ge 0\,,\quad X(t):=\Vert  f(t) \Vert ^{r}_{L^{r}_{ {\ell }}(\mathbb {R}^{d})}\,.\) 
The constant \(\bar{C}_{\gamma }>0\) , defined in (REF ) and (REF ), has been estimated in such theorems as functions of \(C_{\ell , \gamma }\)  for \(\gamma >0\)  or \(C_{\ell ,0}\)  following the definition (REF ).
To this end, integrating the ordinary differential inequality (REF ) proves the following theorem.
Theorem 34 
Let \(b\in L^{1}(\mathbb {S}^{d-1})\) , \(\gamma \in [0,1]\) , \(\ell \ge 0\) , and \(r\in (1,\infty )\) . Assume that \(f_0\)  satisfies that
\(m_{\ell _0}[f_0] <\infty \,, \quad \text{for}\quad \ell _0:=\max \Big \lbrace  1 + \epsilon \,,\,\tfrac{\gamma +\ell }{2} \Big \rbrace \,,\) 
for \(\epsilon >0\)  as small as desired, and that \(\Vert  f_0 \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})}< +\infty \) . Then, the solution \(f(t)\)  of the Boltzmann equation (REF ) with initial datum \(f_0\)  satisfies
\(\sup _{t\ge 0}\Vert  f(t) \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})} \le \max \Big \lbrace \Vert  f_{0} \Vert _{L^{r}_{\ell }(\mathbb {R}^{d})}, \Big (\frac{2\,\bar{C}_{\gamma }}{c_{lb} \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}} \Big )^{\frac{1}{r}}\Big \rbrace \,.\) 
The constant \(\bar{C}_{\gamma }>0\)  is estimated in Theorem REF  equation (REF ) for hard potentials (\(\gamma >0\) ) and Theorem REF  equation (REF ) for Maxwell molecules (\(\gamma >0\) ) . In particular, for the case of Maxwell molecules the argument requires that the entropy must be propagated.
These constants depend on the upper bounds to solutions propagating moments of order \(2\ell \)  for the potential rate \(\alpha \) , characterized in (REF ), associated to the transition probability rate of the Boltzmann flow solved in Theorem REF , were the Lower Bound Lemma REF  calculates the positive constant \(c_{lb}\) , valid for solutions with \(2+\epsilon \)  moments, and defines the coercive constant \(\mathcal {A}_=c_{lb} 2^{\gamma /2}A_\) , with \({A}_{s} =2^{-\frac{\gamma }{2}} c_{\Phi } ( \Vert b\Vert _{L^1(\mathbb {S}^{d-1})}-\mu _{s})\) . In the case Maxwell type of interactions, one can simply take \(c_{lb}=1\) .


Propagation of \(L^{\infty }(\mathbb {R}^d)\) -norm
The main difficulty to overcome in the case \(r=\infty \)  is the fact that the constant \(\bar{C}_{\gamma }\)  in Theorem REF  degenerate in the limit \(r\rightarrow \infty \) . Indeed, recall that Riesz–Thorin interpolation played and important role in the gain of integrability result, in particular, the end-point case \((p,q,r)=(\infty ,1,\infty )\) . This issue is resolved using the following corollary.
Corollary 35 (Upper bound)  Write \(b=b_{S}+b_{R}\)  with \(b_{S}\)  vanishing in the vicinity of \(\lbrace  -1 , 1\rbrace \) . Further, write \(b_{R}(s)= b_{R}(s)1_{\lbrace s\ge 0\rbrace }+b_{R}(s)1_{\lbrace s<0\rbrace }=:b^{+}_{R}(s)+b^{-}_{R}(s)\)  for the forward and backward scattering components supported in \(\mathbb {S}^{d-1}_{\pm }\)  respectively. Then, for any \(\gamma \ge 0\)  it follows that
\(\begin{split}&\Vert  Q^{+}_{|\cdot |^{\gamma },b_{S} }\big ( g , f \big ) \Vert _{L^{\infty }(\mathbb {R}^{d})} \le C_{b_{S}}(2,2) \Vert  g \Vert _{L^{2}_{\gamma }(\mathbb {R}^{d})}\Vert  f \Vert _{L^{2}_{\gamma }(\mathbb {R}^{d})}\,,\\[3pt]&Q^{+}_{|\cdot |^{\gamma },b^{+}_{R} }\big (g , f \big )(v) \le C\Vert b^{+}_{R}\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert _{L^{\infty }(\mathbb {R}^{d})}\Vert  g \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\langle v \rangle ^{\gamma }\,,\quad \text{and}\\[3pt]&Q^{+}_{|\cdot |^{\gamma },b^{-}_{R} }\big (g , f \big )(v) \le C\Vert b^{-}_{R}\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  g \Vert _{L^{\infty }(\mathbb {R}^{d})}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\langle v \rangle ^{\gamma }\,.\end{split}\) 

The first inequality is a particular case of Theorem REF  with \((p,q,r) = (2,2,\infty )\) . For the second just notice, since \(b^{+}_{R}\)  is supported in \([0,1]\) , that \(|u|\le \sqrt{2}\,|v^{\prime }_{*} - v| \le \sqrt{2}\, \langle v^{\prime }_{*} \rangle \langle v \rangle \) . Therefore,
\(Q^{+}_{|\cdot |^{\gamma },b^{+}_{R} }\big (g , f \big )(v) \le 2^{\frac{\gamma }{2}}Q^{+}_{o,b^{+}_{R} }\big (g\langle \cdot \rangle ^{\gamma },f \big )(v)\,\langle v \rangle ^{\gamma }\le C_{b^{+}_{R}}(\infty ,1)\Vert f\Vert _{L^{\infty }(\mathbb {R}^{d})}\Vert  g \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\langle v \rangle ^{\gamma }\,,\) 
where we applied Theorem REF  with \((p,q,r) = (\infty ,1,\infty )\)  in the last inequality. In addition, clearly \(C_{b^{+}_{R}}(\infty ,1)\sim \Vert b^{+}_{R}\Vert _{L^{1}(\mathbb {S}^{d-1})}\) . Similarly, for the term related to \(b^{-}_{R}\) , supported in \([-1,0)\) , it follows that \(|u|\le \sqrt{2}\,|v^{\prime } - v| \le \sqrt{2}\,\langle v^{\prime } \rangle \langle v \rangle \) . Thus,
\(Q^{+}_{|\cdot |^{\gamma },b^{-}_{R} }\big (g , f \big )(v) \le 2^{\frac{\gamma }{2}}Q^{+}_{o,b^{-}_{R} }\big (g,f \langle \cdot \rangle ^{\gamma } \big )(v)\langle v \rangle ^{\gamma }\le C_{b^{-}_{R}}(1,\infty )\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\Vert  g \Vert _{L^{\infty }(\mathbb {R}^{d})}\langle v \rangle ^{\gamma }\,.\) 
Again, Theorem REF  with \((p,q,r) = (1, \infty ,\infty )\)  was applied in the last inequality. The constant is estimated as \(C_{b^{-}_{R}}(1,\infty )\sim \Vert b^{-}_{R}\Vert _{L^{1}(\mathbb {S}^{d-1})}\) .
A direct application of Corollary REF  gives an elementary proof of the \(L^{\infty }\) -norm propagation for the homogeneous Boltzmann equation. As mentioned earlier, a first proof of this fact was given in [2]} under more stringent assumptions. For a discussion with addition of exponential weights refer to [1]}.
Theorem 36 (Propagation of \(L^{\infty }_{\ell }\) -norms) 
Let \(b\in L^{1}(\mathbb {S}^{d-1})\) , \(\gamma \in [0,1]\) , \(\ell \ge 0\) , and assume that for \(\epsilon >0\)  (as small as desired)
\(m_{\ell _0}[f_0] + \Vert  f_{0} \Vert _{L^{\infty }_{\ell }(\mathbb {R}^{d})} =: \bar{C}_{\ell }<\infty \,,\quad \text{for}\quad \ell _0:=\max \Big \lbrace 1+\epsilon ,{\ell +\gamma }\Big \rbrace \,.\) 
Then, there exists a constant \(C^{\infty }_{\ell }:=C^{\infty }(\Vert  b\Vert _1, \bar{C}_{\ell },\gamma ,\ell )>0\)  such that
\(\sup _{t\ge 0}\Vert  f(t,\cdot ) \Vert _{L^{\infty }_{\ell }(\mathbb {R}^{d})} \le C^{\infty }_{\ell }<\infty \,,\) 
for the solution \(f(v,t)\)  of the Boltzmann equation.

Let us consider the case \(\ell =0\)  first. Due to the propagation of mass/energy for the Boltzmann equation,
\(\sup _{t\ge 0} m_{\gamma }[f(t)] < \infty \,.\) 
Consequently \(\Vert  f_0 \Vert _{L^{2}_{\gamma }(\mathbb {R}^{d})}\le \Vert  f_0 \Vert _{L^{1}_{2\gamma }(\mathbb {R}^{d})}^{\frac{1}{2}}\Vert  f_0 \Vert _{L^{\infty }(\mathbb {R}^{d})}^{\frac{1}{2}}<\infty \) , and Theorem REF  applied with \(r=2\)  and \(\ell =\gamma \)  implies that
\(\sup _{t\ge 0}\Vert  f(t,\cdot ) \Vert _{L^{2}_{\gamma }(\mathbb {R}^{d})} < \infty \,.\) 
Now, Corollary REF  (with \(g=f\) ) and Lemma REF  give us that
\(\begin{aligned}\partial _{t}& f (v) = Q^{+}_{|\cdot |^{\gamma },b}(f,f)(v) - f(v) \big (f\ast |\cdot |^{\gamma }\big )(v)\\[2pt]& = Q^{+}_{|\cdot |^{\gamma },b_{S}}(f,f)(v) + Q^{+}_{|\cdot |^{\gamma },b_{R}}(f,f)(v) - f(v)\,\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\,\big (f\ast |\cdot |^{\gamma }\big )(v)\\[2pt]&\hspace{-14.22636pt}\le C_{b_{S}}(2,2) \Vert  f \Vert ^{2}_{L^{2}_{\gamma }(\mathbb {R}^{d})} + C\Vert b_{R}\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert _{L^{\infty }(\mathbb {R}^{d})}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\langle v \rangle ^{\gamma } - c_{lb}\,\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\,f(v)\langle v \rangle ^{\gamma }\,.\end{aligned}\) 
Since \(\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})} = m_{\frac{\gamma }{2}}[f]\le m_{1}[f_0]<\infty \) , we can choose \(b_R>0\)  such that
\(C\Vert  b_{R}\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert  f \Vert _{L^{1}_{\gamma }(\mathbb {R}^{d})}\le \frac{c_{lb}}{2}\,\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\) 
in (REF ) to obtain that
\(\partial _{t}f(v,t) &\le C(b_{S},f_0) + \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Big (\tfrac{c_{lb}}{2}\Vert f(t)\Vert _{L^{\infty }(\mathbb {R}^{d})}\langle v \rangle ^{\gamma } - c_{lb}\,f(v,t)\langle v \rangle ^{\gamma }\Big )\,,\) 

where
\(C(b_{S},f_0):=C_{b_{S}}(2,2)\sup _{t\ge 0}\Vert  f \Vert ^{2}_{L^{2}_{\gamma }(\mathbb {R}^{d})}\le C_{b_{S}}(2,2)\max \Big \lbrace \Vert  f_{0} \Vert _{L^{2}_{\gamma }(\mathbb {R}^{d})}, \Big (\frac{2\,\bar{C}_{\gamma }}{c_{lb} \Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}} \Big )^{\frac{1}{r}}\Big \rbrace \,.\) 
We can integrate estimate (REF ) in \([0,t]\)  to obtain
\(f(v,t) &\le f_{0}(v)e^{-c_{lb}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})} \langle v \rangle ^{\gamma } t} \\&\qquad + \int ^{t}_{0}e^{-c_{lb}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\langle v \rangle ^{\gamma }(t-s)}\Big ( C(b_{S},f_0) + \tfrac{c_{lb}}{2}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\Vert f(s)\Vert _{L^{\infty }(\mathbb {R}^{d})}\Big )\langle v \rangle ^{\gamma } ds\\&\le \Vert f_{0}\Vert _{L^{\infty }(\mathbb {R}^{d})} + \Big (C(b_{S},f_0) + \tfrac{c_{lb}}{2}\,\Vert b\Vert _{L^{1}}\,\sup _{0\le s \le t}\Vert f(s)\Vert _{L^{\infty }(\mathbb {R}^{d})}\Big )\langle v \rangle ^{\gamma }\int ^{t}_{0}e^{-c_{lb}\Vert b\Vert _{L^{1}}\langle v \rangle ^{\gamma }(t-s)} ds\\&\le \Vert f_0\Vert _{L^{\infty }(\mathbb {R}^{d})} + \frac{C(b_{S},f_0)}{c_{lb}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}}+\tfrac{1}{2}\sup _{0\le s \le T}\Vert f(s)\Vert _{L^{\infty }(\mathbb {R}^{d})}\,,\quad \text{ a.e. in } v\in \mathbb {R}^{d}\quad \text{and}\quad 0\le t \le T\,.\) 

The proof for this case is complete after computing the essential supremum in \(v\in \mathbb {R}^{d}\)  and, then, the supremum in \(t\in [0,T]\) . Overall,
\(\sup _{s\ge 0}\Vert f(s)\Vert _{L^{\infty }(\mathbb {R}^{d})} \le 2\Vert f_{0}\Vert _{L^{\infty }(\mathbb {R}^{d})} + \frac{2\,C(b_{S},f_0)}{c_{lb}\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}}\,.\) 
Now, for the case \(\ell \ge 0\)  notice that
\(Q(f,f)(v)\langle v \rangle ^{\ell } \le Q^{+}(g,g)(v) - g(v)\,\Vert b\Vert _{L^{1}(\mathbb {S}^{d-1})}\,\big (f\ast |\cdot |^{\gamma }\big )(v)\,,\qquad g(v):=f(v)\langle v \rangle ^{\ell }\,.\) 
With this observation one can redo the previous proof with \(g\)  instead of \(f\) .

Fine properties of the collision operator
The following computations are reminiscent of [49]}. Recalling the symmetrized weak formulation of the collisional integral, that is, for a suitably regular test function \(\psi (v)\) , the weak form of the collision integral is given by
\(\int _{\mathbb {R}^d} Q(g, f) \psi (v) dv = \int _{\mathbb {R}^d}\int _{\mathbb {R}^d}\int _{\mathbb {S}^{d-1}} g(v) f(w)B(u|, \widehat{u}\cdot \sigma ) \Big (\psi (v^{\prime }) - \psi (v)\Big ) d\sigma dw dv \, .\) 
Let us test in particular with the classical Fourier multiplier,
\( \psi (v) = e^{-i\zeta \cdot v }\) , where \(\zeta \)  is the Fourier variable, we get the Fourier Transform of the collision integral through its weak form
\(\widehat{Q(g,f)}(\zeta ) &=\int _{\mathbb {R}^d} Q(g, f) e^{-i \zeta \cdot v} dv \nonumber \\&= \int _{\mathbb {R}^d}\int _{\mathbb {R}^d}\int _{\mathbb {S}^{d-1}} g(v) f(w) \frac{B(|u|, \widehat{u}\cdot \sigma )}{(\sqrt{2\pi })^d}\Big (e^{-i\zeta \cdot v^{\prime }} - e^{-i \zeta \cdot v}\Big ) d\sigma dw dv \, .\) 

Let us use \(\;\widehat{\cdot }\;\)  to denote the Fourier transform. It should not be a matter of confusion with unit vectors. Assuming a collision kernel \(B(|u|,\widehat{u} \cdot \sigma ) = \Phi (u)\,b(\widehat{u}\cdot \sigma )\) , one has after trivial algebra that
\(\widehat{Q(g,f)}(\zeta ) =\int _{\mathbb {R}^d}\int _{\mathbb {R}^d} g(v)\, f(v - u)\, \Phi (u)\bigg (\int _ {\mathbb {S}^{d-1}} b(\widehat{u}\cdot \sigma ) e^{-i \zeta \cdot v}\Big (e^{-i \frac{1}{2}\zeta \cdot (|u|\sigma - u)} - 1\Big ) d\sigma \bigg ) du dv\,.\) 
Using, in the \(\sigma \) -integration, a reflection that interchanges \(\widehat{u}\)  and \(\widehat{\zeta }\) , it follow that
\(\int _{\mathbb {S}^{d-1}}b(\widehat{u}\cdot \sigma )\Big (e^{-i\frac{1}{2}\zeta \cdot ( |u |\sigma -u)}-1\Big )d\sigma =\int _{\mathbb {S}^{d-1}}b(\widehat{\zeta }\cdot \sigma )\Big ( e^{ i u\cdot \zeta ^{-} }-1 \Big ) d\sigma \,.\) 
As a consequence,
\(\widehat{Q(g,f)}(\zeta ) =\int _ {\mathbb {S}^{d-1}} b(\widehat{\zeta }\cdot \sigma ) \bigg (\int _{\mathbb {R}^d}g(v)e^{-i \zeta \cdot v}\,\int _{\mathbb {R}^d} f(v - u)\, \Phi (u) \Big (e^{ i u \cdot \zeta ^{-}} - 1\Big ) du dv\bigg ) d\sigma \,,\) 
which, in the case of Maxwell molecules, reduces simply to
\(\widehat{Q_{o}(g,f)}(\zeta ) =\int _ {\mathbb {S}^{d-1}} b(\widehat{\zeta }\cdot \sigma ) \Big (\widehat{g}(\zeta ^{+})\widehat{f}(\zeta ^{-}) - g(\zeta )f(0)\Big ) d\sigma .\) 
Equation (REF ) is regarded as Bobylev's formula and has far reaching consequences in the fine analysis of cutoff and non-cutoff collision Boltzmann operators.
Smoothing effects of gain operator
Assume that \(b(\cdot )\)  is smooth and vanishes in the vicinity of \(\lbrace -1,1\rbrace \) . Additionally, assume \(\Phi (u)\)  is smooth vanishing near the origin and at infinity. For any two different unitary vectors \(\widehat{u},\, \widehat{\xi }\) , define \(\varphi \)  as \(\cos (\varphi )=\widehat{u}\,\cdot \,\widehat{\xi }\) . Here we argue in three dimension for the technical simplicity of exposition, however, these ideas can be extended to \(\mathbb {R}^{d}\) , see [50]}, [51]}. In particular see [5]} for the Boltzmann context. In polar coordinates, we parameterize \(\sigma \in \mathbb {S}^{2}\)  as
\(\sigma = \cos (\theta )\widehat{u} + \sin (\theta )\sin (\phi )\widehat{w}_{1} + \sin (\theta )\cos (\phi )\widehat{w}_{2}\,.\) 
Here we choose \(\widehat{w}_{1}\)  in the plane generated by \(\lbrace \widehat{u},\widehat{\xi }\rbrace \) , so, \(\widehat{w}_{2}\)  is perpendicular to both \(\widehat{u}\) , and \(\widehat{\xi }\) . Then,
\(\int _{\mathbb {S}^{2}}b(\widehat{u}\cdot \sigma )e^{-ix\widehat{\xi }\cdot \sigma }d\sigma = \int ^{\pi }_{0}b(\cos (\theta ))e^{ix\cos (\theta )\sin (\varphi )}\sin (\theta )\bigg (\int ^{2\pi }_{0}e^{-ix\sin (\theta )\sin (\varphi )\sin (\phi )}d\phi \bigg )d\theta \,.\) 

The integral inside the parenthesis has stationary phase for \(\phi =\lbrace \pi /2,3\pi /2\rbrace \) . As a consequence, for \(x\gg 1\)  it has the asymptotic behavior
\(\int _{\mathbb {S}^{2}}b(\widehat{u}\cdot \sigma )e^{-ix\widehat{\xi }\cdot \sigma }d\sigma &\approx \sqrt{\frac{2}{x\,\pi }}e^{\pm i \pi /4}\int ^{\pi }_{0}\frac{b(\cos (\theta ))}{\sqrt{\sin (\theta )\sin (\varphi )}}e^{-ix(\cos (\theta )\sin (\varphi )\pm \sin (\theta )\sin (\varphi ))}\sin (\theta )d\theta \\&= \sqrt{\frac{2}{x\,\pi }}e^{\pm i \pi /4}\int ^{\pi }_{0}\frac{b(\cos (\theta ))}{\sqrt{\sin (\theta )\sin (\varphi )}}e^{-ix\cos (\theta \pm \varphi )}\sin (\theta )d\theta \,.\) 

This latter integral has stationary phase for \(\theta =\lbrace \varphi ,-\varphi +\pi \rbrace \) , therefore, we conclude that
\(\int _{\mathbb {S}^{2}}b(\widehat{u}\cdot \sigma )e^{-ix\widehat{\xi }\cdot \sigma }d\sigma \approx \frac{2}{x\pi }b(\pm \cos (\varphi ))e^{\pm i\pi /4}e^{\pm i x}\,,\quad x\gg 1\,.\) 
Now, using this asymptotic expression in formula (REF ) and recalling that \(\Phi (\cdot )\)  vanishes near zero, it follows that for \(|\zeta |\gg 1\) ,
\(\widehat{Q^{+}(g,f)}(\zeta ) &= \int _{\mathbb {R}^3} \bigg (\int _{\mathbb {R}^{3}}g(v)e^{-i \zeta \cdot v/2} f(v - u)e^{-i\zeta \cdot (v-u)/2}dv\bigg )\bigg (\int _{\mathbb {S}^{2}}b(\widehat{u}\cdot \sigma ) \,\,e^{-i \frac{1}{2}(|\zeta ||u|\widehat{\zeta }\cdot \sigma )} d\sigma \bigg )\Phi (u) du\\&\approx \frac{2}{\pi }e^{\pm i \pi /4}\int _{\mathbb {R}^3} \bigg (\int _{\mathbb {R}^{3}}g(v)e^{-i \zeta \cdot v/2} f(v - u)e^{-i\zeta \cdot (v-u)/2}dv\bigg )\frac{b(\pm \widehat{u}\cdot \widehat{\zeta })}{|\zeta ||u|}\Phi (u)e^{\pm i \frac{1}{2}|\zeta ||u|} d u\\&= \frac{2}{|\zeta |\pi }e^{\pm i \pi /4}\int _{\mathbb {R}^{3}} g(v)e^{-i\zeta \cdot v}\, \mathcal {P}_{osc}(\tau _{-v}\,\mathcal {R} f)(\xi )dv\,,\) 

where \(\mathcal {R}\phi (x)=\phi (-x)\)  is the reflexion operator and
\(\mathcal {P}_{osc}(\varphi )(\xi ) = \int _{\mathbb {R}^{3}} \varphi (u)\frac{\Phi (u)}{|u|}\, b(\pm \widehat{u}\cdot \widehat{\zeta }) e^{i \frac{1}{2}(\zeta \cdot u\pm |\zeta ||u|)}\, du\,.\) 
The operator \(\mathcal {P}_{osc}\)  is an oscillatory integral and it is a bounded operator in \(L^{2}(\mathbb {R}^{3})\) , see [50]}, as long as the phase \(S(u,\xi )=u\cdot \zeta \pm |u||\zeta |\)  satisfies
\(\text{Det}\bigg (\frac{\partial ^{2}S}{\partial u_{i}\partial \zeta _{j}}(u,\zeta )\bigg ) = \text{Det}\big (\textbf {1}_{3\times 3} \pm \widehat{u}\,\widehat{\zeta }^{T}\big ) = 1 \pm \widehat{u}\cdot \widehat{\zeta } \ne 0\) 
in the support of \(\Phi (u)\,b(\pm \widehat{u}\cdot \widehat{\zeta })\) . The fact that \(b(\cdot )\)  vanishes near \(\lbrace -1,1\rbrace \)  ensures this condition. Thus, Minkowski's inequality implies that
\(\bigg (\int _{|\zeta |\gg 1}\big |\widehat{Q^{+}(g,f)}&(\zeta )|\zeta |\big |^{2}d\zeta \bigg )^{1/2}\le \frac{2}{\pi }\bigg (\int _{|\zeta |\gg 1}\Big (\int _{\mathbb {R}^{3}}\big |\mathcal {P}_{osc}(\tau _{-v}\mathcal {R}f)(\zeta )\big |\,|g(v)|dv\Big )^{2}d\zeta \bigg )^{1/2}\\&\le \frac{2}{\pi }\int _{\mathbb {R}^{3}}\Big (\int _{|\zeta |\gg 1}\big |\mathcal {P}_{osc}(\tau _{-v}\mathcal {R}f)(\zeta )\big |^{2}d\zeta \Big )^{1/2}\,|g(v)|dv\\&\hspace{-28.45274pt}\le \frac{2}{\pi }\Vert \mathcal {P}_{osc}\Vert _{L^{2}}\int _{\mathbb {R}^{3}}\Vert \tau _{-v}\mathcal {R}f)\Vert _{L^{2}(\mathbb {R}^{3})}\,|g(v)|dv \le \frac{2}{\pi }\Vert \mathcal {P}_{osc}\Vert _{L^{2}}\Vert f\Vert _{L^{2}(\mathbb {R}^{3})}\Vert g\Vert _{L^{1}(\mathbb {R}^{3})}\,.\) 

We know from Theorem REF  that in the region \(|\zeta |\le R\) , for any \(R>0\) , a similar bound holds. This leads to the estimate
\(\Vert Q^{+}(g,f)\Vert _{H^{1}(\mathbb {R}^{3})}\le C_{\Phi ,b}\Vert g\Vert _{L^{1}(\mathbb {R}^{3})}\Vert f\Vert _{L^{2}(\mathbb {R}^{3})}\,,\) 
as long as the aforementioned conditions on \(\Phi \)  and \(b\)  hold. This estimate was first noticed in [5]} and, later, proved by generalised Radon transform methods in [28]}, [7]}. Such conditions can be relaxed a bit, but, such regularisation will not hold for \(b\in L^{1}(\mathbb {S}^{d-1})\) .
This procedure can be generalized to any dimensions and obtain the following theorem.
Theorem 37 
Let \(\Phi \)  and \(b\)  be smooth with \(\Phi \)  vanishing in a vicinity of zero and infinity, and \(b\)  vanishing in a vicinity of \(\lbrace -1,1\rbrace \) . Then,
\(\begin{aligned}\Vert Q^{+}(g,f)\Vert _{H^{\frac{d-1}{2}}(\mathbb {R}^{d})}&\le C^{1}_{\Phi ,b}\Vert g\Vert _{L^{1}(\mathbb {R}^{d})}\Vert f\Vert _{L^{2}(\mathbb {R}^{d})}\,,\\\Vert Q^{+}(g,f)\Vert _{H^{\frac{d-1}{2}}(\mathbb {R}^{d})}&\le C^{2}_{\Phi ,b}\Vert g\Vert _{L^{2}(\mathbb {R}^{d})}\Vert f\Vert _{L^{1}(\mathbb {R}^{d})}\,.\end{aligned}\) 

We end with a remark. Since \(H^{\frac{d-1}{2}}(\mathbb {R}^{d})\subset L^{2d}(\mathbb {R}^{d})\) , then
\(\Vert  Q^{+}(g,f) \Vert _{ L^{2d}(\mathbb {R}^{d}) } \le C \Vert Q^{+}(g,f)\Vert _{H^{\frac{d-1}{2}}(\mathbb {R}^{d})} \le C^{1}_{\Phi ,b}\Vert g\Vert _{L^{1}(\mathbb {R}^{d})}\Vert f\Vert _{L^{2}(\mathbb {R}^{d})}\,,\) 
which provides an interpolation point to prove an analog to Theorem REF  under more stringent assumptions in the angular kernel \(b\) .
Fourier methods have extensive applications in the study of fine properties of the collision operator. These methods have being used to show propagation of Sobolev norms for the homogeneous Boltzmann equation in [7]} under the restriction \(b\in L^{2}(\mathbb {S}^{d-1})\) , based in a result of [29]}. In the work [1]} Fourier methods has being used in full potential to prove propagation of Sobolev regularity for general \(b\in L^{1}(\mathbb {S}^{d-1})\) .

Appendix
The proof of Theorem REF ,
follows the same lines of the argument to solve ODEs in Banach spaces proposed in [60]}. We include it in this manuscript for completeness. The proof is divided into three steps:
Step 1.
First note that since \(\Omega \)  is bounded, there exists a uniform bound \(K_Q\)  of \(Q(u)\) , for all \(u\)  in \(\Omega \) .
Next, let \(u\)  be in \(\Omega \) , then there exists \(h_{u}>0\) , such that
the intersection \(B(u+h Q(u),\epsilon )\cap {\Omega }\backslash \lbrace u+hQ(u)\rbrace \)  is non-empty, for \(0<h<h_{u}\)  and all \(\epsilon >0\)  sufficiently small.
In addition, one can estimate \(\Vert Q(u)-Q(v)\Vert \le \frac{\epsilon }{4}\) , if \(\Vert u-v\Vert \le (K_Q+1)h\) .
Hence, take \(w\)  to be a point inside \(B(u+hQ(u),\epsilon )\cap {\Omega }\backslash \lbrace u+hQ(u)\rbrace  \)  satisfying
\(\Vert w-u-hQ(u)\Vert \le \frac{\epsilon h}{4}.\) 
Next, consider the linear map
\(s\mapsto \rho (s)=u+\frac{s(w-u)}{h},~~~~s\in [0,h], \) 
and see that \(\rho (s)\in \Omega \)  for all \(s\)  in \([0,h]\)  by the convexity of \(\Omega \) .
Moreover, since \(\dot{\rho }(s)=\frac{w-u}{h}\) ,
\(\Vert \dot{\rho }(s)-Q(u)\Vert  \le \frac{\epsilon }{4}.\) 
Now, we can see that
\(\Vert \rho (s)-u\Vert =\left\Vert \frac{s(w-u)}{h}\right\Vert \le \Vert w-u\Vert \le h\Vert Q(u)\Vert +\frac{\epsilon h}{4}<\overline{n}(K_Q+1)h,\) 
which implies
\(\Vert Q({\rho }(s))-Q(u)\Vert \le \frac{\epsilon }{4},~~\forall \, s\in [0,h].\) 
Therefore,
\(\Vert \dot{\rho }(s)-Q({\rho }(s))\Vert \le \frac{\epsilon }{2} \le \epsilon , ~~~\forall \, s\in [0,h]\, ,\) 
and so,
\(\Vert \dot{\rho }(s)\Vert \le 1+\,K_Q ~~~\forall \, s\in [0,h] \ \text{and} \epsilon <1 \, .\) 
Step 2. From Step 1, we have proved the existence of solution \(\rho \)  to the equation (REF ) on an interval \([0,h]\) . From this solution, we carry on the following process.
(1)
We start with the solution \(\rho \) , defined on \([0,h]\)  of (REF ).
(2)
Suppose such solution \(\rho \)  of (REF ) is constructed on \([0,\tau ]\) . Since \(\rho (\tau )\in \Omega \) , by the same process as in Step 1, the solution \(\rho \)  could be extended to \([\tau ,\tau +h_\tau ]\) .
(3)
Next, let \(\rho \)  be constructed by from (REF ) on a series of intervals \([0,\tau _1]\) , \([\tau _1,\tau _2]\) , \(\cdots \) , \([\tau _n,\tau _{n+1}]\) , \(\cdots \) . Moreover, suppose the increasing sequence \(\lbrace \tau _n\rbrace \)  is bounded.
Then, setting
\(\tau =\lim _{n\rightarrow \infty }\tau _n\, ,\) 
then, since \(G({\rho })\)  is bounded by \(C_G\)  on \([\tau _n,\tau _{n+1}]\)  for all \(n\in \mathbb {N},\)  \(\dot{\rho }\)  is bounded by \(\epsilon +C_G\)  on \([0,\tau )\) . Therefore, we can define \(\rho (\tau )\)  satisfying
\(\rho (\tau )=\lim _{n\rightarrow \infty }\rho (\tau _n),\quad \dot{\rho }(\tau )=\lim _{n\rightarrow \infty }\dot{\rho }(\tau _n),\) 
which implies that \(\rho \)  is a solution of (REF ) on \([0,\tau ]\) .

By (3) of this process, we can see that if the solution \(\rho \) , constructed as above, is defined on \([0,T)\) , it could be extended to \([0,T]\) . Suppose that \([0,T]\)  is the maximal closed interval that \(\rho \)  could be constructed, by Step 2 of the process, \(\rho \)  could be extended to a larger interval \([T,T+T_h]\) , which means that \(\rho \)  can be constructed on the whole interval \([0,\infty )\) .
Step 3. Let us now consider two sequences of approximate solutions \(u^\epsilon \) , \(w^\epsilon \) , where \(\epsilon \)  tends to 0. From Step 1 and Step 2, one can see that the time interval \([0,T]\)  can be decomposed into
\(\left(\bigcup _{\gamma }I_\gamma \right)\bigcup \mathfrak {N},\) 
where \(I_\gamma \)  are countably many open intervals and \(\mathfrak {N}\)  is of measure 0.
Taking the derivative of the difference \(\Vert u^\epsilon (t)-w^\epsilon (t)\Vert \)  gives
\(\frac{d}{dt}\Vert u^\epsilon (t)-w^\epsilon (t)\Vert &=&\Big [u^\epsilon -w^\epsilon ,\dot{u}^\epsilon (t)-\dot{w}^\epsilon (t)\Big ]_{-}\\&\le & \Big [u^\epsilon -w^\epsilon ,\dot{u}^\epsilon (t)-\dot{w}^\epsilon (t)\Big ]_{-} + \epsilon \\&\le & L\Vert u^\epsilon (t)-w^\epsilon (t)\Vert  +\epsilon ,\) 
which yields
\(\Vert u^\epsilon (t)-w^\epsilon (t)\Vert  \rightarrow 0 \quad \text{as}\quad \epsilon \rightarrow 0\,,\) 
and we have the convergence \(u^\epsilon \rightarrow u\)  uniformly on \([0,T]\) . The function \(u\)  is, then, a solution of our equation. \(\Box \)

Acknowledgments
The authors would like to thank Ioakeim Ampatzoglou, Erica De La Canal, Antonio Farah, Milana Pavić-Colić and Maja Taskovic for very the careful reading and offering many suggestions that significantly improved the presentation of this manuscript. The authors also thank and gratefully acknowledge the hospitality and support from the Oden Institute of Computational Engineering and Sciences and the University of Texas Austin.
The authors were partially supported by the funding from Bolsa de Produtividade em Pesquisa CNPq (303325/2019-4),
NSF DMS: 2009736 and DOE DE-SC0016283 project Simulation Center for Runaway Electron Avoidance and Mitigation.


